I am learning Simplified Mandarin using Memrise. I just learned the character 觉, to feel. However, I'm pretty sure I learned earlier that 觉 meant to sleep. Is there a difference between the characters?
Chinese characters can have multiple meanings and multiple pronunciations. You figure out the intended pronunciation and meaning based on the context. In the context 睡觉 （Shuìjiào，"go to sleep"), the pronunciation is jiào and the meaning is "(a period of) sleep". In the context 感觉 （Gǎnjué， "feeling"), the pronunciation is jué and it contributes the meaning "feel".
Looking at dictionaries, it seems that the most common meaning of 觉 is to feel or to sense, be aware, etc.
I have only found it related to sleep in the word 睡觉 which means to go to sleep/bed or to sleep, where the character 睡 itself means sleep.
So one can combine the two by assuming something like "to feel (a)sleep", or "to sense sleep", or instead to have your "senses/awareness (falling) asleep". This makes some sense to me and helps me remember, hope it helps you remember too, but please do correct me if I'm wrong here (and judging from the comment by @Lex above saying that jiào et jué are two different words that happen to be written the same, I may very well be wrong).