There are 3 main types of LEDs: miniature LEDs, alphanumeric LEDs, and lighting LEDs.

LEDs are produced in an array of shapes and sizes. The 5 mm cylindrical package (red, fifth from the left) is the most common, estimated at 80% of world production. The color of the plastic lens is often the same as the actual color of light emitted, but not always. For instance, purple plastic is often used for infrared LEDs, and most blue devices have clear housings. There are also LEDs in extremely tiny packages, such as those found on blinkies and on cell phone keypads.

Miniature LEDs
These are mostly single die LEDs used as indicators, and come in various size packages:

surface mount
2 mm
3 mm (T1)
5 mm (T1-3/4)
Other sizes are also available, but less common.
Common package shapes:

Round, dome top
Round, flat top
Rectangular, flat top (often seen in LED bargraph displays)
Triangular or square, flat top
The encapsulation may also be clear or semi opaque to improve contrast and viewing angle.

There are 3 main categories of miniature single die LEDs:

Low current - typically rated for 2 mA at around 2 V (approximately 4 mW consumption).
Standard - 20 mA LEDs at around 2 V (approximately 40 mW) for red, orange, yellow & green, and 20 mA at 4-5 V (approximately 0.1 W) for blue, violet and white.
Ultra high output - 20 mA at approximately 2 V or 4-5 V, designed for viewing in direct sunlight.

Other LED types.

Multicolor LEDs

Bicolor LEDs contain 2 dice of different colors connected back to back, and can produce any of 3 colors. Current flow in one direction produces one color, current in the other direction produces the other color, and bidirectional current produces both colors mixed together.

Tricolor LEDs

Tricolor LEDs contain 2 dice of different colors with a 3 wire connection, available in common anode or common cathode configurations. The most common form of both the bicolor and tricolor LEDs is red/green, producing orange when both colors are powered.

RGB LEDs

RGB LEDs contain red, green and blue emitters, generally using a 4 wire connection with one common (anode or cathode).

5 and 12 volt LEDs

These are miniature LEDs incorporating a series resistor, and may be connected directly to 5 volt or 12 volt power source.

Flashing LEDs

These miniature LEDs flash when connected to 5 V or 12 V. Used as attention seeking indicators where it is desired to avoid the complexity of external electronics.

Alphanumeric LEDs

LED displays are available in 7 segment and starburst format. 7 segment displays handle all numbers and a limited set of letters. Starburst displays can display all letters.
7 segment LED displays were in widespread use in the 1970s and 1980s, but increasing use of LCD displays, with their lower power consumption and greater display flexibility, has reduced the popularity of numeric and alphanumeric LED displays.

Lighting LEDs

LED lamps (also called LED bars or Illuminators) are usually clusters of LEDs in a suitable housing. They come in different shapes, among them the light bulb shape with a large E27 Edison screw and MR16 shape with a bi-pin base. Other models might have a small Edison E14 fitting, GU5.3 (Bipin cap) or GU10 (bayonet socket). This includes low voltage (typically 12 V halogen-like) varieties and replacements for regular AC mains (120-240 V AC) lighting. Currently the latter are less widely available but this is changing rapidly.