I cut my upper arm last night. I didn't think it was that deep, but this morning it is swollen and I can see what I think is fat? Do I need stitches?

No. If you cut it last night and no stitches were placed, it is probably too late to put them in as it is infected by now. I recommend you be seen though as you may need to have it debrided and irrigated and antibiotic started.