The movement is the heart of the watch, the part that actually keeps the time.
Types of movements:
LCD: The most common digital. Uses a Liquid Crystal Display to tell time--Generally black numbers on a silver or gray background, however colors are becoming more common. Some LCD watches have more than just a basic readout of time, such as the Fossil Big Tic series that have animated seconds digits.
Stepper-motor Quartz: The most common type of analog watch. These have one or more stepper motors that move in precise increments. A typical watch will have a single stepper motor to drive the hands through a series of gears, while a chronograph may have one for each independent hand.
LED: The first type of digital watch, using glowing red numbers to read the time. These required a large amount of current for a watch--Typically using 2 of the largest watch batteries available would give a year or so of run time, even with the display only active when the button is pushed. Difficult to read in sunlight. Obsolete soon after the LCD.
Tuning fork--Invented by Bulova, most commonly known as the Accutron. In this watch a tiny tuning fork vibrates at around 300-360 Hz, driving a tiny toothed wheel with a matching number of teeth. This wheel would turn once per second, and in turn drove the rest of the movement. Far more accurate than watches available at the time, surpassed by Quartz which could be made both cheaper and more accurate. Several other watch companies either used Accutron movements or licensed the Accutron patents. Tuning fork watches can be easily recognised by their completely smooth sweeping second hand.
Electric/Elecronic: While these terms would be a technically accurate description for any of the watches above, it is most generally understood to refer to movements with electrically-driven balance wheels. To avoid confusion, these may be known as electromechanical or balance-wheel electric. The earliest versions had no electronic components and suffered from contact arcing which limited the life of the watch. The earliest ones had little practical advantage over automatic watches, and were not particularly reliable.
Hybrid Quartz: Either a balance-wheel electric or a tuning fork movement controlled by a quartz crystal. The first Timex quartz was a balance-wheel electric with a quartz module controlling the rate of the escapement. The first Swiss quartz and the Accutquartz were both tuning-fork based hybrids. A transitional step before switching to stepper-motor technology.
Mechanical: Using the power of a coiled spring to run the watch. Almost all modern mechanical watches use some form of the lever escapement. Some inexpensive watches in the 1800's and early 1900's used a cylinder escapement which is both more fragile and less accurate than a lever escapement.
Lever escapement watches can be divided into pinlever and jeweled-lever, and mechanical watches can be divided into automatic (also known as self-winding) and manual wind. A few watches use a plastic fork shaped like a jeweled lever.
A jeweled lever watch has 2 jewels on the portion of the lever that touches the escape wheel. A jeweled lever movement will almost always have at least 7 jewels--In addition to the jewels on the fork, there will be 2 cap, 2 pivot and an impulse jewel on the escape wheel.
A pinlever watch will have the main body of the fork below the escape wheel, with hardened steel pins extending upward to a specially shaped escape wheel. These watches were generally considerably cheaper, as well as less accurate and less durable over time (although likely more forgiving of abuse) than jeweled lever watches. Typically they would have no jewels, or a single nearly useless cap jewel on the escape wheel.
There were some pinlever watches with a jeweled escapement; these were generally done for marketing reasons rather than for technical merit, since a jeweled lever will help timekeeping far more than a jeweled train.