I had unprotected sex a week ago while I was ovulating. My partner didn't ejaculate. I took a pregnancy test but it came back negative. What's going on?

Sounds normal! In the normal course of events, a man squirts semen into the woman. The sperm in the semen then swim farther up inside the woman to meet the egg, and a baby happens. This means that if the man doesn't do his part, it is unlikely (but not impossible) for the egg to get fertilized.
Too early. One week after ovulation is too early for a pregnancy test, they'll always be negative. Wait until 11-12 days after ovulation, then it should be reliable.
Not sure. You are either not pregnant or you have a false negative test. There are too many factors involved here. If you are trying to become pregnant - it seems like you are since you are having unprotected sex - try again. You might consider repeating the test as the first test might have been done to soon to be positive. If you are not trying to become pregnant, stop playing around.