Don't get confused by the names. Formula One is the world's premier motorsport championship. IT comprises of the world's fastest and most technologically advanced racing cars, combined witht he world's most famous and talented drivers.
A1 Grand Prix is a brand new championship starting this September. It's considered the "world cup of motorsport" as teams will be competing for their country, not for their manufacturers as they do in F1.
All the chassis, engines and tyres in A1 will be uniform and the only thing to seprate the teams will be the technical skill of the race staff and the drivers themselves. F1 on the other hand, is all about technological progress and pushing the limits of design and competitiveness - meaning that the teams with the most resources will be the best.
For someone who doesn't follow motorsport, the cars and their performance will look the same - but for an avid F1 fan the differences are huge and the final conclusion is that A1 is definitely a class below F1 - but this could change if the F1 bureaucracy continue to do shitty amendments like introducing v8 engines instead of the v10s they currently use.