Whether you're buying wheels for style or performance, you definitely want to know what fits. Unfortunately a lot of the online shopping tools are not detailed enough to tell you what wheels really fit. Some might show you wheels that aren't a good choice and others might automatically exclude wheels that would be OK.

1. Will they bolt on? This depends on the year, but for 2001-2006 (Even back to '98, I think), the bolt pattern is 4x114.3 or 4x4.5. That means there's 4 bolts and diagonally they are 4.5 inches (114.3mm) apart. This is not the most common bolt pattern (probably 4x100, or 5-bolt wheels) but it's shared with the Nissan Sentra, Mazda Miata and others. You can search eBay or Craigslist for 4x114 and you'll find wheels that bolt on to your '98-'06 Elantra. Some wheels come with dual 4x114.3 and 4x100 holes - 8 holes means the wheels fit more car models.

2. Offset. Bolting the wheels on doesn't mean they fit right, you need to know if they'll stick out too far (far enough to be cut by the fender edge) or stick in too far (might rub against the suspension). That's where offset comes in. Offset is a measurement of where the wheel is centered relative to the mounting surface, i.e. the brake disc/drum. Wheels with HIGH offsets stick INTO the car, wrapping around the hub and brake components. Wheels with LOW offsets stick OUT more. You can stray from the original spec (+46mm, quite high) around to +45, +43 or even +40 with a little caution. The wheel width matters too - wider wheels leave you less leeway in the offset and most aftermarket wheels will be wider than the stock 5.5" wheels or 6.0" alloys.
In the end, the problem is that there are too many variables as you approach the maximum limits: Wheel width, wheel offset, tire width are easy enough to calculate, but the tire's aspect ratio (and thus the wheel diameter) changes how far the sidewall bulges out. It even varies from tire model to tire model. The best bet is to stay away from the extremes or see what specific combinations people have already tried on Elantras. (link to thread)

3. Weight. If your primary motivation is style, true of most people, weight isn't really a big concern. But if you care about handling limits, you should read a little about the weight of wheels you might consider. Besides changing the weight of the car, wheel mass creates rotational inertia which affects acceleration and braking even more than just dead weight, and wheel+tire weight affects the function of the suspension in an even more subtle way: if you remember your freshman physics - the wheel and spring combination is a harmonic oscillator and its frequency (or time-response) depends on its mass. The lighter the wheel, the faster and better the suspension can do its job of keeping the tires stuck to the road all the time. That's why in racing where handling matters, cars use expensive lightweight wheels. Rota and Centerline make some light (13-15lb) wheels that aren't too expensive, but insisting on light wheels does reduce your style options considerably. Choose your battles.

4. Size. Size is largely up to you. The stock size is 15 inches. If, for some strange reason, you're thinking about smaller wheels... they might not have room for the brake caliper. Even other 15" wheels might have this issue, but most likely you're looking at 16", 17" or larger wheels anyway. Larger wheels are usually heavier (see #3 above), wider (see #2) and more expensive. Also, the larger the wheel, the lower-profile tire you will need to keep the overall diameter the same. Thin tires give you a harsher ride and less cushioning over bumps to protect the wheels. Subject to your tastes, usually 16" or 17" wheels look better than the stock 15" size. Even 18" wheels have been used on the Elantra.

5. Centerbore. The wheel mounting surface is not a simple flat disc - there's raised center part where the axle nut sits inside an opening in the brake rotor. The wheel must have a center bore wide enough to acomodate this feature in the hub. For the elantra, that means a 67.1mm centerbore.