What happened to my soaked almonds? I bought some packed almonds and soaked them in water for about 24 hours. I then removed the water with a sieve, cleaned the bowl and put the almonds back. I covered the bowl with a paper towel and left them for about 48 hours. I then placed them in a plastic container and when I opened it the next day, the almonds were covered with a web-like substance.

Were the almonds infected by fungus? How could this have happened?
They are covered by mold, a type of fungus. What has happened is, mold loves moisture and when you closed the lid on almonds which had a good amount of moisture in them, the mold had the right conditions to grow.

Note: You should discard them all. 

And next time, you'd like to have soaked almonds, it's better to keep them in the fridge at all times, as @rumtscho has aptly remarked. This basically means, you should also soak them in the fridge, and once you strain and discard the water you should continue keeping it in the fridge. Basically treating them as perishable food at all times, and discarding them (if any left ;)) after 5 days...