If you have a male cat, you've probably experienced their territorial rite of spraying. As I was moving out of an apartment many years ago, a neighbor's cat wandered into our place, walked through the rooms, and proceeded to spray a beautiful framed 18x24 photograph my husband had given me one year as a Christmas present. I was quite upset because not only was there no way to fix the damage but I did not think something like this would happen. We did have a cat in heat while we were living here and so her odor, I'm sure, was still spread around. But I had never witnessed this action from a male cat before (I grew up with a male cat in our home but he was an outdoor cat and again, I never saw him spray).
I recently read an article about cats spraying and found that yes, this is a territorial thing, especially if there was a cat in the home (could even have lived there before you). If your cat is having difficulty urinating and is instead missing the litter box, this could indicate another problem. Your cat could have bladder inflamation or cystitis. If the cat is older, arthritis could be making it difficult for your cat to urinate, especially if she is in a lot of pain. As always, consult with your vet if you suspect there may be a problem.