There are many different types of key for sending Morse code (CW) and it can be confusing. If you don't know your cootie from your keyer, or want to understand iambic keying then this is the page for you.
All the different types of key are essentially mechanical, but I am making a distinction between the ones here which can be wired directly into a telegraph or radio circuit ("electro-mechanical") and those that need some supporting electronics.
The simplest and oldest way to send Morse is through a "straight" or "telegraph" key. A straight key is essentially just an electro-mechanical switch with a big lever: at rest the switch is open but when pressed the switch is closed. A straight key would normally be wired directly to a radio transmitter (or telegraph wire) so that the transmitter is turned on when the switch is closed. As the timing of the Morse is controlled entirely by the operator, each element (dit, dah, space) may be timed slightly differently, with an operator's individual style known as their "fist". The up and down motion of the straight key when used over a significant period could cause a repetitive strain injury known as "glass arm" and hence all the more recent keys use a side to side motion.
A "cootie" or "sideswiper" is a bit like a straight key but turned on its side so that the motion is not up and down but rather side to side. However, a cootie is at rest in the middle (where the switch is open) but can be pushed either to the left or to the right (where the switch is closed). Pushing left or right completes the same circuit and when using a cootie, the operator alternates between one side and then the other. As with the straight key, the operator is in full control of the timing and will have an idiosyncratic style, influenced by the alternating motion of the mechanism. A cootie has no electronics and would be wired directly into the radio transmitter.
A "bug" or "semi-automatic" key is the most complex electro-mechanical key. It has the advantage over the cootie in that it assists the operator in sending well-formed streams of dits. As with the cootie, the bug can be pushed to the left or to the right. Pushing a bug's lever to the left (conventionally) makes it act just like a cootie (as a manual key) with the timing being under the operator's control. Generally, pushing to the left was only used for sending dahs because the key feature of a bug is that pushing it to the right and holding it there sets off a mechanical device which repeatedly sends dits (with intra-character spacing). Whether sending manual dahs or automatic dits, the bug completes the same circuit and would be wired directly into a transmitter.
A "sounder" was used in the days of sending Morse by telegraph wire and hence would have been used with American (Railroad) Morse and a straight key. the sounder converted the pulses of electricity coming down the wire into audible sound: not the electronic beep-beep that is produced by a keyer's sidetone today but a mechanical click and clunk sound. The sounder works by pulling a sprung metal bar down with an electromagnet when current was present (causing a "click") and then the bar would be released when the current turned off (causing a "clunk"). The "click" is therefore heard when a dit or dah starts and the "clunk" at the end of a dit or dah (hear an example). A sounder and a straight key were often mounted on the same base and known as "key on base".
As noted above, the keys themselves are not necessarily electronic, but they require electronics to operate.
A "single-lever paddle" (with a keyer) is the modern electronic version of the bug, though mechanically it is more like a cootie. Compared to a cootie, the difference is that a single-lever paddle has two separate switches/circuits: one for left and one for right. The operator positions the paddle between their thumb and finger and pushes one way or the other. A paddle is not wired directly to a transmitter, instead it must be wired to a "keyer" (see below) which uses electronics to automatically create a series of dits for one side and a series of dahs for the other.
A "dual-lever paddle" (or "squeeze key" or, somewhat incorrectly, "iambic paddle") is two paddles positioned right next to one another, each operating a separate switch. As with the single-lever paddle, the operator positions the paddles between thumb and finger. A push to the right will push the left lever to the right and a push to the left will push the right lever to the left. In this way, when wired up to a keyer it is no different to the single-lever paddle, but the advantage is that with a modern "iambic" keyer you can get a useful effect by activating both paddles at once - squeezing them together. When used with an iambic keyer the result of squeezing the paddles together is a string of alternating dits and dahs.
The "keyer" converts the on/off electrical signals from the single or dual-lever paddles into dits and dahs. A keyer can be a separate piece of equipment or can be integrated into a radio transmitter. Either way its function is the same: it will have a way to adjust the speed in words-per-minute and may have some more advanced timing configuration. Using the speed setting the keyer will produce well-timed dits when one paddle is activated and dahs when the other paddle is pressed. In a sense it is the modern-day equivalent to the mechanical bug which mechanically produced a series of dits. A keyer may also have an oscillator and speaker built in. The sound produced by this is called the "sidetone" and a keyer with sidetone may be used for Morse practice without connecting to or activating a radio transmitter.
Modern keyers all support "iambic" mode when used with dual-lever paddles. When iambic mode is chosen on the keyer and the paddles of a dual-lever paddle are squeezed together, the keyer will send an alternating series of dits and dahs which comes in handy for the (English) letters CFKLQRY (and "ä", the opposite of "C"). The word "iambic" is used as it refers to the "iamb" or a combination of an unstressed and a stressed sylable in English poetry. For instance in Shakespeare's Sonnet 12: "When I do count the clock that tells the time" or "di-dah di-dah di-dah di-dah di-dah". The alternating sequence starts with a dit or a dah depending on which paddle is pressed first. If just one paddle of the two is released then the keyer will continue sending just dits or dahs depending on which paddle remains active. If both paddles are released simultaneously then there's a further complication: in "iambic A" mode the keyer will stop sending, but in "iambic B" mode the keyer will add on one more element to the alternating sequence. Confused? There's a great video demonstration you can watch showing a single-lever paddle, dual-lever paddle and iambic A and B modes.