What does this (electrical) value mean on my washing machine's back?
The symbol to the left is the IEEE (Institute of Electrical and Electronics Engineers) symbol for a fuse, so it's telling you you need a 10A fuse.  A 10A fuse will allow a maximum sustained current of 10A; anything above that will cause the fuse to blow -- how quickly it does will depend on the current that is passing through the fuse.

See the Wikipedia page for electrical fuses for more information