The simplest answer is that an ileostomy is an opening for waste and an urostomy is an opening for urine. The ileostomy is a simpler construction though - the colon (and sometimes the rectum as well, but not always) is removed and the end of the small bowel is brought out through the stomach and formed into a stoma, a posh Greek word for 'opening'.
With the urostomy, a stoma is also formed. A short segment of small bowel is cut out and formed into a tube to which the two ureters from the kidney are sewn into. One end of this tube (called an ileal conduit) is sewn up, the other end is brought out through the stomach and formed into a stoma. Urine passes through that into a bag.
You might hear the term colostomy occasionally. A colostomy bag is what people who don't have a bag invariably call it, but half the time they're likely to be wrong. An ileostomy is formed from the ileum, the end of the small bowel, whereas as a colostomy is formed from the colon - in the latter case, the entire colon hasn't been removed, obviously. Anyway, just thought I'd bung that in.
Hope this helps! Diagrams are helpful if you can't visualise it. Google Image search is your friend (or not, depending on how squeamish you are >_>)
Hi NiceCupOfTea, Thanks for the info.