Terraria is a fairly simple game. You start off with nothing, so you chop down a tree, build a house, then go out and mine some ore. You make armor out of that ore so that you can brave underground caves and fight off monsters to discover chests with unique items in each. Eventually, you'll be strong enough to summon and fight boss monsters, who drop better and better loot (and many of them unlock harder content upon their demise!)
I think by far my favorite game mechanic is the general sense of progression. I like the feeling of going through a game, getting powerful items, then returning to older stuff and being able to repeat that content far more easily. This is one of the core aspects of RPG's, where you level up and grow inherently stronger simply by playing the game. In Terraria, however, it's a little different. There is no leveling system, so one's ability to clear challenging content is determined by skill and previously obtained equipment.
In this game specifically, there is also a strong element of randomness. Each world is randomly generated, and while certain things are always in every world, some things are not. One of the best examples is the pyramids one can sometimes find almost completely buried in the generated deserts. These structures are rare, and only around one in ten worlds will have them. They aren't necessary to fully experience and appreciate the game, but things like this make the idea of going back and making a new character a year later far more attractive.
There's a lot this game has to offer regarding how you want to fight, too. First, when you're creating a character, you can choose 'softcore', 'mediumcore', and 'hardcore', which determines what happens when that character dies. Softcore characters drop their money, mediumcore characters drop everything they currently have in their inventory, and hardcore characters die for good. If you're playing this game for the first time, even if you think you know what you're doing, choose softcore. You'll thank me later. But beyond that, when you're creating a world, you can choose either Normal or Expert mode. The difference is that Expert mode has a lot of murderous death monsters added to the world, and the payoff for that is unique items that specific monsters drop. Beyond this, though, when you beat the 'final boss' in the game, "Surprise!" It wasn't the final boss. You have now entered 'Hardmode'. The world now naturally spawns harder enemies (especially if you're playing in an Expert world) and there are now several new bosses to fight. When you beat that boss, you finish the first 'half' of the game.
But the game isn't solely focused around combat. You can enjoy yourself immensely just by playing a softcore character on a normal world, ignoring every boss. That's because another thing that this game does very well is the ability for players to do what they want in the game. There are over two dozen non-player characters that can 'join' your town (if it's large enough) and over two hundred different types of blocks means you can build whatever your heart desires. You can build huge cities, tall towers, or underground cathedrals. I've tried building huge monuments, and the finished product makes it worth it, but I'm not usually patient enough to take the time to build something awesome. It's definitely the reason a lot of people play the game, though. While they don't quite match up to the cities and structures teams of people have taken the time to build in Minecraft, there is certainly something to be said for the creativity people bring into it.
Overall I think the best way to play this game is to find a few friends, play together on the same server, and have everybody play the way they want to. What I've found is that most people pick up 'jobs' and just go in that direction. One or two people will build the city, one person will go out and explore for hidden dungeons and temples, and one person may just be the 'get all the materials' guy. Sounds boring, but that's actually the role I find myself in most often. Going out and supplying everyone with what they need will make everybody like you.
So the game is a lot of fun, and it lends itself to a huge sense of wonder and discovery regardless of how many times you've played it before. Unfortunately this is one of those games that you can't simply go on YouTube and watch somebody else play it. In order to enjoy it, you have to play it yourself. But trust me, it's worth it.