The American cheese we know today is the quintessential American product ... cheap, processed, convenient. Still good when melted on a burger, patty melt or a fish sandwich. Other cheeses have more flavor, but completely different textures that don't always stack well on a greasy hockey puck on a white bun.
American cheese was probably derived from Monterrey Jack, which is another fairly bland, but real, cheese. I think if you buy most "jack" cheeses today, like the cheap pepper jacks common in grocery stores, it's just another variation on processed cheese. In the US we have a remarkable selection of really _bad_ cheese. From the packaged pre-sliced stuff and the monster sized blocks in the dairy section, to the garbage they sell in most grocery deli cases.
Velveeta is much the same thing as American, except even softer and more meltable. I understand Velveeta is available in some places outside the US. So soft that it's hard to slice, so isn't used much on sandwiches - usually just melted for things like nachos, queso dip and mac & cheese.