The four most common methods:
Straight Key: The classic up-and-down lever. You make every dot and dash manually. Simple, classic, relatively inexpensive and it teaches you proper timing. No batteries or power supply, can key almost anything. Requires more effort than other methods and most ops' top speed is somewhere between 15 and 25 wpm.
Semiautomatic Key or Bug: Classic side-to-side mechanical key. Thumb side makes dits automatically, finger side makes dahs manually. Requires no batteries or power supply, can key almost anything, no external boxes needed. Speed adjustment is mechanical, most bugs cannot go slower than 15-18 wpm unless extra weights are added. Requires less effort than a straight key and offers more control than a keyer.
Electronic Keyer: Electronic side-to-side key. Thumb side makes dits automatically, finger side makes dahs automatically. Requires power supply. May require special interface to key various rigs, particularly a newer keyer mated to an older rig. Speed adjustment is electronic, range depends on design. Many modern rigs have a keyer built in, and require only a set of paddles. Requires less effort than a straight key but not as much control as a bug.
There are several types of paddles and keyers in modern use.
Paddles come in two basic styles:
Single-lever, in which the key can be pushed one way for dits, the other way for dahs, and left alone for spaces.
Dual-lever, in which the dit and dah sides are independent, and both can be closed, or only one, or neither.
With a single-lever paddle, what you have is essentially a fully-automated electronic bug key, where the dahs are made for you too.
With a dual-lever paddle, other options arise, depending on how the electronics behave when both levers are closed. When you see "Iambic Mode A" and "Iambic Mode B", that's what they mean. The various modes have their advantages and disadvantages, which is a discussion in itself.
Keyboard: Not a key in the usual sense, but a typewriter keyboard that generates Morse Code. Operator does not have to know the code or develop new skills. Usual implementations are the following:
1) A computer equipped with Morse Code generating software is used to convert keystrokes to Morse Code. This is by far the most common implementation, but of course requires a computer, interface, and software.
2) Stand-alone special-purpose Morse keyboards have been made, and were somewhat popular before computers became inexpensive and ubiquitous. Now somewhat unusual.
3) A few rigs can generate Morse Code directly from a typical computer keyboard plugged into the rig.
IMHO the best way to learn is to start with a straight key and get used to it, as it teaches timing and a feel for the code. Learning to send helps one learn to receive, too.
73 de Jim, N2EY