Keyboards in the traditional physical sense come in basically two types as far as the actual type of keys used. This then usually determines all the other defining characteristics of the keyboards and different options.

Mechanical

Mechanical switch keyboards are often larger full-size height key types. They tend to be louder and maybe what you traditionally think of as a keyboard with a desktop computer. This area of keyboards is where the hobbyists are, not only are there so many options available but it's possible to assemble your own keyboards.

The defining characterising is that underneath the key it is a physical switch that is pushed down to make a contact.

Membrane

These are the more flat key type keyboards and tend to be more silent and with far less travel of your fingers to actually type and press a key. They are the type of keyboards you will most often associate with those built into laptops.

Rather than a physical push mechanical switch, these often have a plastic sheet (membrane) covering the entire layout of keys and the action of pushing on this sheet in one place (a key) brings it into contact underneath and activates a keypress.

Hybrid

Most other keyboard types are actually some sort of cross between a membrane and mechanical type key press. The most common will be Apple's keyboards that have never had a full membrane-type but they are not actually vertical push buttons either.

The current type of key press technology Apple uses is a scissor-switch which doesn't have a membrane but keeps a much lower profile than a traditional mechanical switch.