"35mm," refers to the film it takes (there are other film formats, but 35mm (135) is the easiest to obtain and process (turn exposed film into images).
Most cameras these days are digital - the film is replaced by a sensor, and images are stored on a removable card - while the starting price can be a bit steep, you only have to buy the odd card and keep the camera in batteries (most cameras come with a rechargeable battery in any case) so running costs are much lower.
For your first digital camera, just look around for something that fits your budget, avoiding the obviously bad like Vivitar, Kodak, GE, Samsung and BenQ. Canon and Nikon tend to be expensive for what you get in this category.
My first digital camera was a Casio compact, small enough to slip into a pocket, but a better performer than it had any right to be for the price.