I haven't been to birthday parties of some age groups, but here is what I know.
Babies: Not exactly the birthday, but there is a celebration when the baby turns 100 days old. The baby is taken to a shrine to be blessed, some nice food is prepared, and photos taken after dressing the baby with a silk dress.
Birthday parties for babies, held at home, are usually small events with just the family members. Cake shops provide free name plates and candles for babies and small kids.
Daycare centers and Kindergartens hold monthly birthday parties, for all the kids born in that month. A cake, songs, and also greeting cards with photos are included in these events.
Kids at primary school might invite a few friends to a small party at home.
University students have a variety of birthday parties, just like anywhere else. Surprise parties in parks, restaurants, and also on-campus, are not uncommon.
Some restaurants provide special services for birthday dinners, particularly for the youth. Songs and music, cakes, and even fortune tellers! I once received a surprise birthday dinner with a fortune teller. She had a positive forecast, I guess this is the case for anyone :-p.
For working dads, small parties or dinners might be held on the nearest weekend. I guess this depends on the family.
For elders, there are special birthday parties when they are 77 or 88 years old. Many get invited, and those invited also get a gift from the family.
This is what I have seen. @Japanesemina might be able to give us more interesting details.