Are some infinities bigger than others?

Yes. In the 19th century, Georg Cantor proved that the real numbers, all the numbers with endless decimals, are more numerous than the counting numbers 1, 2, 3... even though both collections are infinite. In fact, his method implies there's an infinity of different infinities.

The way to compare infinite sets is to pair them up one-to-one. Some infinities that look different turn out to be the same size. Imagine a hotel with infinitely many rooms, all full. A new guest arrives? Ask every guest to move from room n to room n+1: there's no last room, so everyone fits, and room 1 is free. Infinitely many new guests? Move everyone from room n to room 2n, and all the odd rooms open up.

But some crowds never fit. Suppose you claim to have listed every infinite string of 0s and 1s. Cantor builds a new string by flipping the first digit of the first string, the second digit of the second, and so on down the diagonal. The new string differs from every string on your list somewhere, so it isn't on the list. No list can ever be complete: that infinity is bigger.

The idea was so shocking that a fellow mathematician, Leopold Kronecker, called Cantor a "corrupter of youth".