Hi, My dd is now 5 but I have a list of what she got on her 1st bday and I remember her favorites...besides the stacking cups that were already mentioned, she got a Fisher Price Little People Cinderella set, with the carriage with horse and some figures--and she absolutely LOVED that. The Leap Frog ABC caterpillar has been a favorite of visiting children; my dd liked dancing to its music but otherwise didn't really play with it, so I guess it depends on the child.
She also got balls (larger ones) and books, which I highly recommend--they are a favorite for a long time. Bath books and touchy feely type books are great. It probably goes without saying, but at this age they should not be watching any tv or videos (even educational stuff--it messes with their brain and is actually bad for them until age 2). Music is great, though--Raffi, Veggie Tales, Taj Mahal, Baby Einstein (she liked the Baby Vivaldi a lot) etc. I took her to Kindermusik classes and I highly recommend them for overall fun as well as age appropriate stimulation--you could ask for the tuition for a gift (it's about $56 for 8 weeks I think).
Things she got at 18 months (Christmas) that were favorites were Duplos, calculator/phone/beeper (toy versions), a little plastic chair/stool and table, puzzles, and dolls and a little kitchen (all STILL played with today).
I personally prefer non-electronic toys and things that are classics, they seem to go further and last longer--but some electronic things certainly add to the fun.
She typically loved pulling my tupperware out of my cabinet (I left one without the safety latch for stuff that she could play with, and she never bothered the latched ones) or put things in and out of boxes/purses/bags (like unloading the dryer in to the basket or pulling little things like old credit cards or keys out of my purse), and chewing on crinkly paper or cloth. So any "gift" that was along those lines was a hit...like a shape sorter for example.
At the party itself she had her first ear infection so she wasn't feeling too well but she really enjoyed the coconut pudding and sour cream & onion potato chips (go figure!).
Hope that helps, it's getting more fun isn't it!?
Lynn