A public server ("pub") is where you play ET all the time, pretty much. If you're in a clan, you might play on your clan's server, (or on the server of the opposing team). But for the most part, pubs are where you play.
There are a few things to consider:
Ping
Less than 50 is desirable. 50-80 is okay. 80-140 is playable. Above 140 and
it really gets difficult.
Settings
How many people are playing?
If you are new to the game, then the best server is probably one where lots
of people are playing, i.e. 24 or more people. This will give you a chance to
learn the game and play around, without having to worry about being a crucial
part of the team.
If you have some experience playing ET, probably the best games are from 12 to
26 people. That way you have enough people that the map isn't boring, but not so
many that the server gets laggy, people start crowding the doorways, panzers run
around and TK 10 guys at once, etc. Many "elite" players prefer games even
smaller, like 12-16 people.
Of course, it's a matter of preference, you should
just try out different servers and see what you like.
What's the map rotation?
Do you want to stick to the classic 6 maps or are you looking to try some
new maps? If you haven't ever tried anything but the stock maps, I suggest you
try 'em. A lot of them are of really good quality, or even conversions from RTCW
maps. Some servers give you the best of both worlds by including say, 3 stock
maps and 3 new maps in the rotation.
How good are the players who play there?
This is a very important factor. Although every
server has its mixture of noobs and good players, the skill level on pubs varies
a LOT. Try to find a server
which is suited to your skill level.
If you're on a server and you're getting beat up by everyone, find a different server. It's no fun for you to get beat up, and it's no challenge to the other players.
On the other hand, if you're the one beating up everyone on the server, leave. You won't be improving your skills because it's no challenge, and they'll feel bad because they keep dying. If you must play on a noob server, at least try to play a class you don't usually play so you can give them a fighting chance. If you always play on a noob server, you'll think you're really good, but then the moment you try your tactics out on some better players, you'll get whooped.
So how do you go about finding a good server, other than trying every single one that appears in the server listing? Well, you could try going on one of the ET forums and searching its archives to see what servers people have suggested in the past. (This question has been asked before, trust me!)
Another good way to find good servers is to use ASE ("All-Seeing Eye"). It's a program which lets you browse the list of all servers in ET and see who is playing on them. Look at the player list: if all the players have white names, or if none of the players have clan tags, then it's probably a noob server. That's not to say that all colored-named people or all clanners are good players, but there is some truth to it. Get to know the names and tags of some clans which represent your skill level. For example, if you play against a few guys from clan XYZ and they seem pretty good, then you could search for their clan tag to find what servers they are playing on, and then just play on those servers. Usually people within a given clan are of roughly equal skill level.
I actually recommend using ASE over the in-game ET browser. The ET browser is kind of buggy from what I have seen, and also ASE is nice because you can just click on a server and instantly see who is playing there. What I do is I just add all the servers I've ever played on to my favorites. This way when I want to refresh my server list, it will happen very fast, because I only play on maybe 20 different servers, as opposed to the thousands of ET servers which actually exist.
Which mod is the server running?
Personally I find this the least important factor. If it has low ping and
good players I'll play on it. If you want to know more about mods though, the
major two are etpro and shrub, and I'll talk about them in the next section.
One thing to keep in mind though is that shrub servers tend to be less
competitive than etpro. So if you want to play against some good players, it may
be tougher to find a good shrub server than it is to find a good etpro server.
etpro and shrub are the two major variations (mods) of ET. shrub is very popular on pubs, and offers lots of customizations. etpro is meant more for competitive play so you'll use it in clan matches, although it is also popular on pubs. Many people who are in clans hate playing on shrub, but personally I think both etpro and shrub have their advantages. Here's a comparison:
etpro
shrub
etpro is great if you are competing, since it's basically what's used in all competition play. The main problem I have with etpro is that there aren't any nifty text shortcuts like there are in shrub. This isn't an issue in clans because you'll be using TeamSpeak or Ventrilo, but on pubs using etpro, I miss having the text shortcuts. However, with the new etpro 3.0.0 that's been released, I like etpro the best now because it has hit sounds. Hit sounds are incredibly useful because now you can fire across the map and know if you're hitting anything. Also, if you're an MG, you can fire through smoke or through trees and you'll know if you're hitting stuff. I guess this is kind of a nasty exploit but I like it.
shrub is cool because it has so many customizations. Goomba kills rock, and playing dead is a pretty cool feature. The text shortcuts are also very handy. The bad thing about shrub is that with a shrub server, you never know what you're going to get, because everything is so ultra-customizeable. For example, there's one server I used to visit occasionally where all the charge bars would recharge 10 times faster. So, panzers could shoot all their 4 shots practically one after the other. Field ops could toss constant airstrikes. And the worst is that covert ops would just run around throwing smoke cannisters all across the map, so the entire map would be covered in smoke and you couldn't see anything.
Most servers aren't that extreme though. For the most part, you could join a server and you wouldn't know if it was etpro or shrub, until you tried playing dead or something like that. (Playing dead only works on shrub).
The other unfortunate thing about shrub is that there are a fair number of bugs. For example the, "revive yourself when you die" bug, or the "walk through walls" bug.