There's this really nice comment from one user that explains it really easily
Imagine you would have a magic book. There are people around the world who also have a copy of the magic book. Now, everyone can write into that book; if you write something into the book, it appears in all other copies of the book, around the world, pretty much instantly. Nobody, not even you, can erase what you wrote into that book or what someone else wrote into that book. That's basically blockchain technology, on which Bitcoin, Ethereum, and all other cryptocurrencies are based.
Bitcoin, for example, uses the magic book to log transactions of the virtual currency "Bitcoin". You'd basically write into the magic book "I give Cathy 1 Bitcoin". The book then checks through all the entries of any entries of me having Bitcoin in the first place and if so, send the intended 1 to Cathy. Now Cathy has an entry in the magic book and now it knows she has some Bitcoin too, so if she tries to send one, the magic book can confirm (thanks /u/Kroucher for the suggestion).
Ethereum goes further, here you can basically write small "self-executing" programs that can't be deleted. So with our book analogy, you could write into the book "I bet 1 ETH with Amy that the weather on May the 2nd is sunny and nice", and include a data link to Google Weather or similar which the program can check. You pay a small fee to deploy the program. The code then keeps getting executed on thousands of computers worldwide. If it actually rains on May the 2nd, the program sends 1 ETH from you to Amy's wallet. Since anyone can check the code of the program, Amy doesn't have to trust me, the creator of the program, she can check my code for herself. Also, the program can't be suddenly deleted (so I can't just quit the bet after looking at the forecast on May the 1st).
So Ethereum is basically a giant, coordinated decentralized swarm of individual's computers connected to each other on the web where anybody can deploy his mini-programs (called smart contracts) for a small fee.
It's slow tho