What Do Servers Offer?
In Terraria, servers are the life of everything in the multiplayer arena. You can play single player without having to get into one, but if you want to join other players in a larger open world to roam, you'll need to join a server. So what do Terraria servers offer for you that you can't get on your own? The answer is: Plenty! There are a number of benefits to be had for joining a mulitplayer server.
Companions. Multiplayer servers are just that; multiplayer. Some will be low-population and give you only a couple of additional players to party with or compete against. Others will be higher population servers with tons of players roaming the world, doing their own thing. Other players can benefit you in many ways; they can help you build, they can help get you started, they can give you surplus items you're looking for, and they can help kill difficult monsters when you're stuck on your own. Also, who doesn't need a little extra help piecing together a building, tower, cave, or other structure you want to make? More hands makes the work go faster.
Settings. Servers allow many custom settings, everything from higher monster spawns and different layouts of worlds, to established towns, extra NPCs, and for the more customized out there, nearly anything else. The number of settings you can change is huge, and the custom servers out there can be nearly infinite. Make the game easier, or more challenging, or just more fun with wacky add-ons.
Rules. Some servers you may join might be full of horrible people, digging death traps under spawn points, yelling racial slurs like it's Xbox Live, and generally being horrible people. Other servers may have stricter requirements, and ban people who grief, swear, or otherwise make themselves into nuisances. One of the worst things that can happen to a community is the people in it turning sour, and good rules and active moderators can help keep a server fresh, interesting, and fun for people who play in it. Of course you're going to find the occasional server of horrible people, but those are few and far between.
Communities. Some servers only allow people from certain communities to join them. Whether you're joining one based on a particular state or region of the US, or a brit-only server, or a server that only allows people with the last name starting with S, the server you join can be a huge difference in the player base. Anything is possible!