So you want to make 3D art? Well it's actually pretty easy to start, as there are plenty of free programs, and tutorials out there. But first there are some basic terms and concepts you should be aware of. The first is what makes a 3D model, and that is:
Polygons - Polygons are the flat planes that 3D models are made of. I'm sure you've seen them in their faceted glory on older game consoles like the Playstation, or older PC games. Each polygon is made of points called vertices. Vertices are connected by edges. Edges are connected by faces. Like so:
Polygons: Usually nicer looking than this.
Each vertex has a location in space determined by 3 dimensional coordinates, X, Y, and Z. Y is usually up and down, X and Z are usually side-to-side and front to back respectively (though this can be different and is best to be aware of). The 3 dimensional grid they are placed in starts at 0 in the center of the scene. You don't normally need to know their exact position, but it you should be familiar with how it all goes down. For example, if you have a vertex at -1,2,-1, and you're facing down positive Z, It would be 1 space to the right, 2 up in the air, and 1 space behind the center of the grid.
Pictures are easier.
If you need an easy way to remember what direction X,Y and Z go, hold your right hand out in a fist. Extend your index finger and thumb. Now hold your middle finger perpedicular to your index finger. Your thumb is +Y, you index finger is +Z, and your middle finger is +X
Many polygons are connected together to form the model (also called a mesh). Polygons can have 3 to infinite edges. However it's best that you keep them at at 3 or 4. Many programs won't even import models with 5 or more sides. Meshes can be colored with textures, but you'll need a model finished before you start worrying about that.
Now you need a means of manipulating these verts (short for vertices), which brings us to the programs. There are plenty of free ones out there you can use, but if you have the cash I like Maya. 3DS Max is also good, though I'm not too familiar with it. If you're interested in sculpting, zBrush is the industry standard. Anyway, here's a list of free programs which you'll probably want to get started with anyway so you don't spend 3000 dollars on a program you end up hating.
Blender – All around great program. Does it all, modeling, uving, rendering, rigging, animation, even has a game engine built in. Probably the best free 3D package out there. Though it takes a bit of getting used to.
Wings3D – Great for just getting right to it. It's a low poly modeler, it can uv, it's really easy to just pick up and play with. Small file size. Recommended for first timers.
Sculptris – If you're interested in the 3D sculpting of organic models, this is the one for you. It automatically adds detail where you need it. It's tons of fun in a 2 mb file. I actually recommended it for anyone whether you're interested or not. It's like digital Play Doh
The GIMP – Not a 3D package, but if you plan on texturing your models at all you'll need a program to do that with. Last I saw Photoshop was still 300 dollars, so this is the best free alternative.
I recommend playing around with these programs, and learning what you can from practice. Hope this helped you get started. More to come.