Lasting from the mid-50s through the 60s, The Silver Age is the product of the Comics Code Authority. If you are a long-time comics fan you've probably heard of the CCA, but if not then here's the basic rundown.
Once upon a time, a man named Dr Fredric Wertham performed a terribly done and thoroughly debunked study from which he concluded that comics were giving kids the gay and making them juvenile delinquents. In response, comics wound up censoring a truly incredible list of things, including gore, swearing, disrespect for authority and even things like slang. If your comic met all the required criteria you got a stamp from the CCA and people would buy and sell your book.
Now, some of these things actually did need to be regulated, but the restrictions were so extreme that you couldn't really do much of… anything. Certainly the days of realistic and nuanced discussion were behind comics at this point, in no small part because the CCA banned showing criminals in any kind of sympathetic light.
Because of this, comics kicked off the goofy nonsense age that so many comic fans remember with such disdain and revulsion. Personally, I love the Silver Age and I'm glad the CCA came along to give it to us. Sure, it cut out some great stuff, but we have lots of serious comics before and after it, and the pure unadulterated camp can be great fun. As a reference here, think about the 1960s Batman TV show, with the bright colors, dancing Batman and constant humor.
There were some downsides to the Silver Age however - notably women took a real step back in these comics. In the Golden Age women might not have been done terribly well, but they were often front and center and in the thick of the action. Daring reporters like Lois Lane, superheroines like Wonder Woman and Black Canary and a variety of super-villainesses weren't too hard to find.
Then there's the silver age. As part of the 60s' backlash to women pushing gender roles in the 40s America whiplashed into strict gender roles again, and comics was dragged down with it. Romantic interests suddenly spent far less time chasing out danger on their own (or at all, in many cases), there was a drastic reduction in independent female heroes and villainesses practically disappeared.
What female heroes were around tended to be side characters - Supergirl, Batwoman, Batgirl and so forth - and it's worth noting that the original Batwoman and Batgirl (Kathy and Betty Kane) were nothing more than stock romantic interests for Batman and Robin (this is before Batwoman was Batman's cousin and also gay).
While some female villains existed - one notable villainess creation was Poison Ivy - their appearances tanked to almost nothing.
The Justice League only had one representative of the fairer sex - Wonder Woman - for most if not all of the Silver Age. Black Canary joined in 1969, during the transition to the Bronze Age. To make things worse Wonder Woman left the JLA in the late 60s to be a super-spy, so for a period the Justice League had no women at all. As noted last chapter, even the Justice Society of America wound up with more than one woman before they ended their run. It is also worth noting that the JSA's female characters were Wonder Woman and Black Canary as well, making the JLA even more derivative.
The Silver Age, similar to the Golden Age before it had some other hallmarks. One is that heroes still don't argue with each other. The Justice League is such a picture of idyllic harmony that even readers in the 60s were put off and wrote in about it! If you read the letter pages you can see readers repeatedly ask why the Leaguers never seem to argue - and they always get the same answer: the Justice League represents idealized heroes who, of course, would never disagree.
The Silver Age is also when gimmicked supervillains really start hitting their stride. The Silver Age is largely considered to begin with the introduction of the Barry Allen Flash in 1956 (the Comics Code Authority began in 1954 so there is a grey area as to when the Silver Age truly begins). Appropriately enough, Barry Allen would go on to embody the Silver Age.
His Rogues Gallery is made up of extremely gimmicky supervillains. His powers have a heavy emphasis on pseudo-science and you can find many an amusing 'scientific' explanations for the happenings within his comic. There are practically no useful female characters in his comic - Iris barely does anything plot relevant and Barry's only major female villain, Golden Glider, would not be introduced until the Modern Age. Flash got in on the sidekick craze of the 60s as well, with the introduction of Kid Flash. If you want a comic that really hits every traditional note of the Silver Age, Flash is it. And you can read his crossovers with Green Lantern!
The Flash/Lantern friendship is the second of its kind in comics, only coming in behind Superman/Batman. While all heroes are marginally and automatically friends, only Superman/Batman and Flash/Green Lantern had established friendships with each other in particular with regular appearances in each other's comics, and references to their friendships even coming up in the Justice League of America comic!
Recommendations
Flash - For all the reasons mentioned earlier, Flash can be a great look into the Silver Age. He's got all the hallmarks of a Silver Age comic - for better or for worse - but they're usually pretty well executed. Silver Age Flash introduces the first multiverse by bringing back the original Flash, Jay Garrick, and Kid Flash can be fun as well. Also, despite not being a terribly useful character, Iris West actually becomes a pretty supportive romantic interest by Silver Age standards - I may write a chapter on her at some point too. If you like some of the hallmarks of the Silver Age but wish the writing was generally better, then good news - Barry Allen functionally stays in the silver age for a really long time. His 70s comics are still in the Silver Age in a lot of ways, so that might be a good balance for you to strike.
Teen Titans - The 60s heralds the debut of everyone's favorite teenybopper crimebusters and Daddy-o is it a wild ride! Teen Titans is fascinating because you can really see it trying to find its identity early on, and the tone fluctuates wildly on occasion. Unfortunately, the Silver Age aspect of the Teen Titans doesn't last for terribly long since they start hitting the Modern Age relatively soon. However, if you want to read an easily manageable amount of comics, then this might be a feature rather than a bug.
While other comics may be more well-rounded than the Teen Titans, I can think of no other series that distills the Silver Age in the way Teen Titans does. As kid heroes they may have a group personality like the grown-ups, but that personality is dynamic to the max. They speak almost exclusively in comically exaggerated slang the writers thought was how kids actually talked. Their missions were ridiculous, and the gimmicks they used in return equally ridiculous. Every issue is a rollercoaster and I cannot recommend it enough for some pure fun.
Also, by some sheer coincidence Teen Titans actually hit on actual good female representation in the 60s. Wonder Girl's only personality trait may be boy-crazy teenybopper but to be fair that is one more distinguishing personality trait than everyone else gets, so that's something. More to the point, Wonder Girl is the big gun of the Teen Titans, and the comic actually treats her like it. While there are many powerful female characters on teams, they're rarely the big gun even today - sure Wonder Woman is powerful on the JLA, but the big gun is Superman, for instance. Not all teams even have one character who is clearly the big gun - but Teen Titans did. In their first issue the Teen Titans effectively face four monsters and a big bad - WG is part or all of the solution to taking out three of the monsters and the ultimate mastermind - talk about being the MVP! Teen Titans also introduces DC's first black hero - Mal Duncan, and black female hero - Bumblebee.