If you are doing what is generally termed "recreational" Judo, you should be fine. There is always a possibility of injury to your fingers/hands/wrists/arms regardless, it's a contact activity.
Several people have mentioned collarbone fractures -- when you're thrown forward such as by seoinage in particular, resist the temptation to put your hand straight out in front of you down on the mat.
Elbows, of course. And over the long term, knees and shoulders. Donn Draeger did a whole load of different MA's but always said that judo was hardest on the body. In fact he called judo "the great crippler." He was very tough, lots of weight training, but also very flexible, something he emphasized for injury prevention.
Damn, that's wretched. His videos are universally terrible. I pity his students. Even the Israeli Ninjas do it better:
Uki waza, done properly, is the first set in this video
Op: Judo is awesome, but its not for wimps.
Uki waza used to be one of my main throws for BJJ. Never did it that beautifully but it was effective. Now I find I rely more on kososto gari/gake, kouchigari, and Uchimata.
Just got Mallet Finger on my L ring finger from judo training :[
This is where the tendon that straightens the tip of the finger has ruptured and fails to straighten all the way.
It wasn't sore, but 6 weeks in a splint!!! 6 weeks out of action (Plus 6 hours in A&E, but thats another story!)
Judo is tough! The common skeletal injuries are broken fingers or toes. You are going to get sprains, bruises and strains.
Toe/finger sprains aren't uncommon, fractured a toe myself.
There are 3 primary causes of injuries in Judo:
1 - Ego
2 - Forcing/reaching for what isn't there (works closely with above)
3 - White Belts
Learn your ukemi (both sides!), don't worry about getting thrown, don't force what you don't have (cause of my ACL injury), and keep a close eye on that big new guy who hasn't been around long enough to learn 1, 2 and 3.