i managed someow to launch and use Terraria on a mediocre Toshiba netbook that used a lame single-core intel Atom 1.5GHz, had 1 Gb of ram, and used HD graphics... running windows 7. its ridiculous really; the operative system alone pretty much consumed half of the PC's processing power. and it STILL worked!
... if you can whitstand 20 minutes of waiting for a new SMALL map to be created, 3 minutes for it to load, playing on a screen with about as much resolution as a Gameboy Advance, lagging terribly with more than 4 light sources on-screen, and being unable to play on large maps at all.
truth is, terraria is so insanely light for today's standarts (the game occupies what, 400 MB?) that it pretty much comes as a guaranteed bet for any computer you use.
personaly, since my high school course is about animation and i will need a good rig for processing, i invested on a very, VERY solid asus N551j that has a ludicrously overpowered processor (i7 quad core 3.5GHz with 8 threads? YES PLEASE!), a hefty load with 12 GBs of ram and some freaking 4GB dedicated to graphics with the small but very efficient Geforce GTX850M. with this rig, i can play terraria on insane settings (pffft.... i can play Far Cry 4 maxed, Skyrim maxed with 4K textures from mods, i even dabble in Crysis 3 on mid settings with around 30-40 FPS... this computer is simply overpowered), with everything maxed on, and a resolution so massive i cannot even figure out what colour are the eyes of my character anymore. that is, if you are willing to spend 1300€ on a laptop that tends to enjoy overheating... seriously, dont use this laptop without a cooler pad.
but for a "playable" experience with affordable costs, 500€ should do just fine. make sure you have an intel i3 at the very least (a mid-high i5 is just enough for max settings already), ANYTHING that is not the integrated HD graphics (no matter... i dont care if you pick Nvidia or Radeon, anything will be preferable) and has the highest DirectX compatibility (9 is pretty comon and lets you play 90% of the games out there by the way) and at least 500Mb-1GB dedicated graphcs. i recall seeing some Geforce 500-600 class for about 80€ that has these specifications, shouldnt be too hard to find. it is enough to play a lot of games on its own, too. all the way up to left 4 dead 2... after that, it starts to be too obsolete. for ram, 2GB are more than enough, but if you plan on using large maps, high resolutions, or like to build very complex stuff with dozens of different tiles in use, maybe 4GB is a safer bet.
in Terraria's case, you should focus more on a processor than on graphics. terraria has lightweight graphics. but the true wheight of the game comes from the Scripts that run in the background - the game engine handling massive spawns, deciding random drops and loot, keeping track of everything you store, saving the location of every single block in the game, controlling the growth rate of plants... in many aspects, you can consider it a trimmed down version of Minecraft in what respects Spec demands. not a lot going on in the graphics department, but should you be careless you lag your system with too many scripts at once. this happens a lot if you run the game on a Pc that nears it's limit of power, and activate a long and complex set of Redstone. its not that the Graphics are failing - the processor + ram are, instead. in terraria, this is very visible when you summon multiple bosses in a room full of automated traps - tons of different AI scripts, timers, levers, teleports, moving player hitboxes like crazy... it really chews up your GHz.
the screen you pick should be proportional to the graphics power. too much screen resolution for a weaker graphic card means that you cant play fullscreen without performance loss, which kills the experience.
last bo no least, youj should pick an efficient Operative system. DO NOT USE VISTA. that mumbliong abomination is the worst optimized thing i have ever seen. windows 7 is the way to go. if you feel unsafe, Windows XP is ridiculously light and fully compatible with terraria, and WILL improve your performance, but mind the security risk, lack of support and new drives, and low compatibility to most things in the internet today.
if you are bold, and ESPECIALLY if you use gameiki mod, i would suggest you something that probably NO ONE ever suggested...
a wacom graphics tablet.
i am not joking! it actually works with the game... i once even killed the Wall of fless with Orichalcum-tier armor and weps without breaking a sweat.
but the wacom is inferior to the mouse for killing things, obviously.
however, if you use Gameiki extendively, and use a lot of the Creator mode with long reach enabled, you will be SHOCKED at how handy this digital pen can be - you literaly end up building things in terraria as if you were spriting on MS Paint! it is like painting, really, and allowed me already to create very interesting things.
it just so happens that this is a very unique way of playing... i do that sometimes. this comes more as a personal suggestion for fun than for actually playing the game. oh, and wacom tablets cost above 100€, so unless you plan on learning photoshop, dont even consider the investment.