Overview
One of the Three factions of the Citadel Council, flatly ignoring the various 'minor races' that the council itself seems fond of ignoring that is. The oldest race probably deserves a thorough write up so lets start with them first.
Biology
The Asari are a monogendered race, meaning that without exception there is only female Asari. Their body plan overall resembles that of a human female with few noticeable differences, firstly being the fringes sticking out of the back of their heads in place of hair, and the second being their skin tones typically ranging from Teal, Navy and Purple tones. Details about their bodies that aren't immediately visible include a life span approximately 1000 years on average, a strong affinity for the often bizarre gravity manipulating Mass Effect phenomena due to a extremely high percentage of Eezo in their bodies and finally a bizarre method of reproduction in which they psionically link their nervous system to another's and use that to map out the other person's DNA onto a blank Embryo being stored inside of their ovaries. To be clear: this process can be done supposedly with anyone of any species and doesn't require physical contact to happen. Yet to confirm rumours.
Asari carry a bizarre social stigma against 'Purebloods' or Asari/Asari pairings, originally the explanation given about this was that it does nothing to advance the race's genetic diversity, instead it has come to light that there is a rare neurological disease that appears among the Ten Percent of children of Asari/Asari pairings, however the likelihood of this Neurological disorder appearing shoots up all the way to eighty percent if one of the Parents is a Pureblood herself. The Ardat Yakshi are probably worth an article by themselves but to break it down into brass tacks: the Neurological disorder has three symptoms. Firstly the Asari in question are born sterile, No Exceptions. Secondly, with the exception of a synthetic partner, the normal bonding process with a Ardat Yakshi results in immediate total neural burnout, in other words death by massive brain haemorrhage. Thirdly, the bonding process completely drains the target of all their skills and experience and gives them to the Ardat Yakshi in question which also enhances their innate psionic abilities in a highly addictive process, meaning that the Ardat Yakshi themselves are more inclined to repeat the mating process. If you think that these various features are abnormal and unnatural you're probably right but it's better covered in history which leads to...
History
Pre-space flight Asari history is pot-marked by a prolonged period under the thumb of the Protheans who spent a considerable amount of time and resources engineering them to have a number of features, total Biotic ability and relatively high average psi rating being the least of them. Because of the Asari's unique reproductive system as long as they have access to another organic creature with a brain then they can have a child, this means that even if they get exterminated almost down to the last Asari they can still rebuild their race from scratch as many times as needed. It is unknown if the Asari were meant to serve as exceptional front-line combatants during the last cycle or rulers of this one, whatever plans the Protheans had for them likely died with the Protheans themselves. There is a religion set up dedicated to the worship of the Prothean overseer responsible for looking after the aliens, Athame, whether Athame herself set up the religion or it is simply a unintended side effect is unknown at this point in time. After Athame came the Asari equivalent of the medieval ages, various clans and city states ruled by Ardat Yakshi who were treated as goddesses of sex, death and destruction. Even though this is described as Medieval level culture they had made it all the way through an industrial society resulting in a massive rebellion against the Ardat Yakshi with the invention of their equivalent of rifles. Also because my colleagues are pigs and insist I point this out anyway the above Ardat Yakshi condition also means that the Asari themselves probably had to resort to bestiality and literal man-hunting to survive. I have to live with these images in my head and now so do you.
Among the Citadel government the Asari hold the distinction of being the first race to reach the Citadel, narrowly beating out the Salarian's in the process, and when a third race, the Volus made their presence known the first two decided to form the Citadel Council and exclude the physically, technologically and politically weaker Volus. This would be impressive up until you realise that the Prothean's heavy interaction with the Asari meant that they had more access to intact Prothean technology than anyone else in the galaxy, and continued to monopolise this technology even with the threat of the Reapers and the Union on the horizon, and even after narrowly surviving two other wars and in more recent years the cold war with the Geth still didn't share this technology. Course that isn't the worst of the Secrets the Asari chose to hide from the rest of the Galaxy, the existence of the Ardat Yakshi was covered up and written off as religion and superstition with the religious military order the Justicar's being founded to police anymore Ardat Yakshi that appears in the future.
During a age of rapid expansion the Asari met the various 'minor races' of the galaxy and gradually awed them into the fold of the Citadel Council, this also inevitably led to first contact between the Rachni and the Asari's partners in the Salarians and the Rachni wars that followed. The Rachni was the Council's first real trial, an implacable race of insects with seemingly limitless numbers and tomb world preference meant that not only were they a massive threat to the Council's corner of the Galaxy but also they couldn't get to them to deal any real damage to them, then the introduction of the Krogan to the Galaxy managed to solve the matter. And then the Krogan Rebellions happened in a glorious display of why the Stellar Union does not uplift species. Once again on the back foot the Asari apparently did relatively little until the Turian's came into the galactic fold and through the use of the Salarian engineered bio-weapon known as the Genophage sent the Krogan on a downward spiral to extinction. Apparently not yet done with starting wars and then not participating in them the Asari also observed that another race in the Quarians had been working to circumvent laws banning AI through the Geth and bluntly told them to stop, which then led to the Morning war, the Quarian's exile into space and politically severing ties to the new established Migrant fleet while they waited for a solution to dealing with the Geth to come up.
Military
Looking at the above you might notice a lack of any real military action on the Asari's part, aside from battles that either consisted of special infiltration or defensive, which in of itself speaks of the Asari's military Doctrine. Observing that most races cannot meet their 1000 year long life span the Asari are reluctant to cut their long life spans short by engaging in military action preferring to wait for their enemies to be defeated as opposed to doing so themselves. While the Asari understand the need for armed forces, and in the wake of back to back wars even appreciate them, the Asari ultimately see the Military as just that, an unfortunate necessity. Given what we know about the Protheans and what their plans for the Asari likely were we can safely assuming that any remaining Prothean corpses have spent the past several thousand years rolling in their graves. However one should not mistake a lack of inclination towards violence as being bad at it, on the contrary Asari place a hyper fixation on quality over quantity therefore what forces that they do have are some of the best trained in the galaxy armed with the finest weapons. However this hyper fixation on Quality over Quantity has backfired on them in the past. The fight against the Rachni depleted stores of weapons that had previously taken the Asari centuries to amass, again not because the Asari are bad at making them but at the time they placed a emphasis on Quality over Quantity to the point that until that point in time they were still making their weapons by hand.
Needless to say, the Asari quickly adopted Salarian techniques for mass production of weapons, though production of ships remained in the hands of dedicated guilds and families if the Asari are going to become useful during the incoming Reaper war this policy will need to see change as well. Regardless, onto the subject of Asari ground forces. Asari can join the military as 'young' as 100 years old, where they are then sorted into one of three categories depending on how they specked their Biotic implants when they were a teenager.
* Commando Adepts-General purpose Biotic specialists with a general emphasis on enemy harassment techniques, they're job generally being either holding the enemy in place while someone else takes the shot or immobilising them while the rest of the unit beats a hasty retreat.
* Commando Sentinels-Tech specialists and tanks on the battlefield. Through the use of specialised high precision biotic training and implants the Sentinels are able to mold Omni-gel into floating Omni-Armour suspended in their own biotic field, generally on the job for taking hits. Also generally good hackers and tech experts taking on the majority of the work when it comes to stealing high value information.
* Commando Vanguards-Highly trained Asari warriors with a emphasis on raw damage out put and hit and run techniques, opposite of usual Asari doctrine of 'wait and see' or 'stay out of sight' the Vanguard's job is to apply the Asari's own considerable biotic ability to breaking enemy lines, generally charging into the enemies face, unloading a shot gun or machine gun of some kind, and then retreating.
Over all the Above three specialist positions create fearsome combinations of biotic mastery providing support, cover and offensive power to break a enemy unit over the knee, with the introduction of Union technology to the mix the overall fighting power of these three units should see an impressive hike, and yet despite this the Asari government has continued it's emphasis on quality with their special forces units, the Huntresses.
* Huntress: Imagine if you would, you were part of a race whose life span stretched several centuries, what would you do for a special forces soldier? Well the Asari's response was to run the Huntress through the ringer of all three classes of training, Adepts, Sentinels and Vanguard's all rolled into one. You would get a Unit capable of Charging around the battlefield armoured like a Krogan Linebacker lining up the unfortunate mooks that get in their way with a variety of immobilising techniques, as if that wasn't terrifying enough the Asari government also saw fit to give these monsters invisibility as well. Because of course they did. For certain high ranking Asari officials the use of a unit of Asari Huntresses has been a pretty good threat. Until recently that is...
Lessus was something of a embarrassment for the Asari government, the Ardat Yakshi was the dirtiest among their dirty secrets, on the other hand it's discovery by the Union is one of the biggest boons that the Asari as a whole has received. Negotiating a similar deal as the Krogan the Asari government has made a simple arrangement, training for their people in psionics, advance cybernetic augmentations and three different Tactical Gear exo-armours, in exchange said augmented and trained units will aid in dealing with threats to the Union, chiefly the Prethoryn at the moment, and the reapers in the future under the Justicar Cabal.
The Asari Naval power is few and far between, for the bulk of the fighting they have Turians, for stealth and infiltration the Salarians, what few ships they do keep is best reflected in their flag ship, the Destiny Ascendant. The Ship in question is one part mistake, one part carrot and one part stick. First and foremost was the mistake, when the Drive Core that would become the central power unit was made a clerical error had spat the damned thing out at twice the size it should've been, and so the ship had to be scaled up to compensate. Secondly was the stick, because the ship was so hideously large and powerful it could also be strapped with enough fire power to reduce entire solar systems to so much space debris floating in the solar winds. Third the carrot, the Destiny Ascendant is the biggest luxury resort in council space, because even after you dedicate all of the surface space outside of the ship to fire power the inside was basically empty. Thus the massive heavily armed luxury cruiser is filled with spa's, Strip clubs, Brothels, resorts, even artificial beaches complete with an artificial lake that can reportedly be surfed. The concept is simple 'scare primitive barely space capable civilisations into submission, then welcome them into the lap of luxury' and the laps of nubile Asari strippers and whores. The Asari don't keep war ships, they keep tools to perpetuate their political agenda