An almost ring is like a toy box with some toys missing. Normally, a toy box would have all kinds of toys, but in an almost ring, there might be some toys missing.
In the same way, a ring normally has things called "multiplicative inverses" for every number, which is a fancy way of saying that each number in the ring has a buddy that they can multiply with to get a special number called "1". But in an almost ring, some of the numbers might not have a buddy to multiply with to get 1, just like some of the toys might be missing from the toy box.
It's still a type of ring, but it's a bit different because not all the numbers have buddies for multiplication. Sometimes, mathematicians study almost rings because they can still be useful even if they're not "perfect" like regular rings.