In regard to the landing itself, there's two pretty famous errors: Apollo 11 missed its targeted area by between 3-4 miles, and it had more fuel than the instruments indicated. For a good synopsis of these two errors, check out this transcript of the Apollo 11 radio transmissions, with some great annotation by Armstrong and Aldrin.

There are lots of theories for why Apollo 11 missed its target, but some of the more prominent ones involve the undocking procedure from the command module (air from the docking tunnel pushed the two craft apart) and the landing module's water boiler, which gave off some vapors (and later caused problems for Apollo 13).

As for the fuel difficulties, it turned out that fuel in Apollo 11's LM tanks sloshed around more than predicted. The fuel tanks had some baffles to reduce the sloshing, but it continued to a greater-than-expected extent. Future fuel tanks had more baffles to eliminate the problem.

Yes! The landing itself was tricky because it was the only part of the mission that hadn't been rehearsed in earlier flights. Perhaps the most famous were the program alarms (1201, 1202) generated by a problem with the way radar data was processed by the Apollo guidance computer. (Warning: am historian, not comp sci) A program alarm during powered descent was a big deal- well, really any problem with the computer. Turns out the computer was just telling them that it had some bogus data and it was marking it low priority and dropping the jobs out of the queue. Mission Control figured this out and gave the astronauts the go to continue their descent.

Another problem was the condition of the landing site, which Neil could see out the window. It was covered in huge boulders, so he used up most of his fuel scooting around looking for a smooth patch to land. At bingo fuel they would have had to abort the landing. Obviously they parked it in the nick of time.

I'm about to go to bed, but I think there might be a couple more interesting stories I can add after a look at the ol bookshelf, and I can add sources too.

Edit: You can read a blow-by-blow account of the landing in James Hansen's First Man: The Life of Neil Armstrong. The portion about the program alarms begins on page 458 (and goes on for about 6 pages). If you want to learn about how the Apollo computer was built and operated, I highly recommend David Mindell's Digital Apollo: Human and Machine in Spaceflight. Chapter 9 describes the landing and details the alarms. Mindell's account is much more technical and than Hansen's and describes the other moon landings as well.

http://www.cnbc.com/id/42592372

A circuit breaker in the lunar lander broke, so Aldrin had to jimmy it with a pen in order to arm the ascent engine.