The Euler method as applied to games is pretty much just the naive
Position + (Velocity * TimeSinceLastUpdate) formula I think you were getting at. The other methods (without getting into the calc too much) are just more accurate ways of estimating velocity and position based on multiple simultaneous forces, like friction, gravity, air density, etc. It's probably most noticeable in accurate flight simulators, where the aircraft actually needs to behave according to different forces across the wings' control surfaces.
So, referring back to the Euler method, an integral is essentially a repeated sum.
Position += Velocity * TimeSinceLastUpdate; // for every frame!
Which in graph form might look like
where the width of each rectangle is the time, and the height is the velocity. Obviously we can get better estimations the more rectangles we use, but that means doing more processing each frame. If we can do the job of several integrations in one step by, say, using sloped rectangles, then our physics will be just that much more accurate.
Of course, this isn't always necessary - accurate isn't always interesting.