"Profile" is a a pretty meaningless word there, actually. It just means that wave moves through space! I suppose it is trying to get at what the wave looks like from side on (in profile), like a cross-section through the wavefronts.
A standing wave is e.g. on a guitar string. The standing wave on the string makes the string oscillate, but the wave remains where it is i.e. on the guitar, because it's held down to zero displacement at both ends.
A travelling wave is e.g. on a lake when you throw a stone in. The stone makes the inital displacement to start the wave, but then the wave moves off away from its source. If waves didn't travel, this would be a disaster because then light from the Sun would not be able to reach us, making life quite difficult.