A gearless bike is a one gear bike, called a "fixie". A fixie is good only if all your riding will be on the level. If you intend to ride on even mild hills, you'll be pushing a lot.
A geared bike has various means to keep the chain taut enough, the main one being a jockey wheel on a spring-loaded arm to equalise the tension over the various chain lengths resulting from the gear (tooth) combinations coming into play. The chain comes off if you are clumsy with the gears, but you'll soon learn to adjust the chain and gears, and to change them, so as to minimise chain dismounts.
At your budget, you'll probably be better off to buy a good quality used bike rather than a poor quality new bike.