What is the best flooring material to use in a basement that sometimes gets wet?

Trying to avoid linoleum or polished cement. Trying for a more contemporary, elegant look.

Tile or flagstone.

Wall to wall sponge. Just pull it out and wring it out when the floods occurr.
Seriously, you may want to think about fixing what is causing the flooding. IF that is not feasable, there are some good commercial floor products which will withstand getting wet but will still look good. You could try tile, perhaps that terra cotta tile which is used in restaurants. Most McDonalds have this stuff in them if you want to go see how it looks. Another type of flooring could be one of the poured-in-place epoxies. There is one type with colored flakes in it which looks great. There are others which have a more industrial look, which is usually compatable with a "contemporary" style.
I wish you well, wet basements are no fun at all.

It would be advisable to find the source and fix the problem with the basement getting wet before installing any floor covering. You will end up with a mold problem under the flooring material.

I recently installed wood laminate in my basement that I purchased from IKEA. Pergo is the most common name brand. It looks great and is designed for wet places. It can be sealed around the edges to be waterproof.

get the garage floor paint that has a sealer in it. It will not only be functional but will look good

Check out this website below. I added this flooring in my basement at my old house and loved it.

There is product you can get a home depot that is a subfloor material on top with a plastic honey comb pattern layer on the bottom. it snaps together and allows water to run underneith without allowing water up onto the flooring. You can put whatever kind of flooring ontop of it but you might want to stay away from hardwood if there is too much moisture, might cause it to swell.

Two types of material come to mind. Epoxy paint or terrazzo. The paint would be less expensive but you would have to deal with fumes for a week or two. Terrazzo floors are made of marble and epoxy and are very expensive.