Shown below are the most common types used at most universities and research facilities.
If you feel none of them suit your particular application, let me know and we can look into other specialty options.

Standard Taper Ground Joints

Used to connect glass apparatus and/or components together. Most frequently used with a sealing grease applied between the two (inner and outer) joints, though thin teflon sleeves or teflon rings are available and can be used in place of the grease. Poor seals and stuck (frozen) joints are often the result of using connectors "dry" or improperly greased.

To determine size:
EX. 24/40 .... The first number is the approximate OD of the inner joint at the largest end. The second number represents the approximate length of the ground portion of the joint.

Spherical (Ball and Socket) Joints

Used to connect glass apparatus and/or components together. Unlike the fixed and rigid standard taper joint, the ball and socket joint design allows limited side to side movement. This flexibility can be helpful in some glass systems where alignment is a problem. Most ball and socket joints are used with grease, but versions are available that incorporate the use of O-Rings or teflon coatings to aid in sealing. A pinch clamp is used to hold the ball and socket together.

18/9 ... The first number represents the approximate OD of the ball. The second number is the ID of the tubing.

Threaded Connectors

Threaded connectors offer an alternative to the standard taper ground joint. The greaseless seal is made by O-Ring compression using threaded connectors with internal or external threads. The versatility of this type of connection is useful to the laboratory without quick or easy access to a professional glassblower. Can be used to make connections of dissimilar materials.

Size references ( # 7, 11, 15, etc.) may indicate different measuring points depending on the manufacturer and the type of thread. These connections have a range, not a set size, partially determined by the size of the O-Ring used. Note connector threads are not necessarily the same as cap threads. It is suggested that the manufacturers technical data be referenced for additional information.

O-Ring Joints

This style of connecting joint works well when assembling components within a limited space. The seal is made by O-Ring compression using a pinch clamp. The glass joint can be used grease-less, though some will lightly coat the O-Ring with grease in the belief they will get a better seal. O-Ring material (of which there are many) selection offers wide range application of this connector.

The size # referenced is usually the ID of the joint/tubing. # 25 is an O-Ring joint with a 25 mm opening or ID.

Glass Stopper (Flask) Joints

Used on volumetric flasks, bottles, etc.. Shorter ground portion (length) than found on most standard taper joints. Most stopper plugs are solid.

Size #'s approximate the OD of the plug at its largest point.

Glass Plug Stopcock

Standard glass valve used for non-vacuum systems and apparatus. The glass plug is usually solid, exposed at the upper and lower ends of the taper, and retained in the glass barrel with a clip or washer. Plugs are standardized and interchangeable within their size. Used with grease. Stopcocks size is determined by the diameter of the hole (bore) through the plug. A
4mm stopcock has a hole 4 mm in diameter through the plug ( not to be confused with the inner diameter of the glass tube attached to the stopcock barrel).

Note: Glass stopcock plugs cannot be used in glass barrels designed for teflon plugs. It is advisable to visually check all used glass plugs prior to using, confirming plug bore and alignment with openings in the barrel.

Teflon Plug Stopcock

Standard glass valve used for non-vacuum systems and apparatus where grease cannot be used. The teflon plug is exposed at the top and bottom of the taper, retained in the barrel with a teflon washer, o-ring and nut (in that order). Note the barrel of teflon plug stopcocks is smooth and polished, unlike the ground finish found in glass plug stopcocks.

Stopcock size is determined by the diameter of the hole through the plug - not the ID of the tubing attached to the stopcock barrel.
Note: Teflon plugs cannot be used in glass “hi-vac” barrels.

Glass Hi-Vacuum Stopcocks

Unlike standard non-vacuum stopcocks, most hi-vacuum valves minimize potential leakage by leaving only one end exposed to the atmosphere. The lower sidearm may come off a bulb as shown in this example, or it may come off of the barrel itself. In either case, the lower sidearm should always face the source of the vacuum. These valves are designed to be used with hi-vacuum grease. Stopcock size is determined by the diameter of the hole passing through the plug.

Glass Hi-Vacuum Stopcocks - Plug Design #1

The plug shown is used with the barrel pictured above. Glass hi-vacuum plugs are usually hollow, using vacuum as an aid in setting the plug. If the stopcock is installed incorrectly it would not be possible to "pump down" the plug through the bottom opening. This design exposes the internal portion of the plug to the system.

Glass Hi-Vacuum Stopcocks - Plug Design #2

This stopcock design gives the user a few more options in retaining plug vacuum when opening the system to atmosphere. The small hole in the plug, directly below the plug "bore", is aligned with the lower sidearm (vacuum source) and evacuated. The photograph does not show it but the "bore" is actually a tube passing through the plug. This feature isolates the system from the vacuum held in the plug.

Glass Hi-Vacuum Stopcocks - Numbers

All glass hi-vacuum stopcock plugs and barrels are numbered and matched. Ground to tighter tolerances than standard stopcocks, plugs and barrels are not interchangeable if you want to achieve hi-vacuum.

Hi-Vacuum Stopcocks - O-Ring

O-Ring style vacuum valves are found in systems that cannot tolerate exposure to stopcock grease. Available with different o-ring and barrel composition, exposure and configuration, this style of valve has found wide-spread use. Unlike glass hi-vacuum stopcocks, the barrels are interchangeable, making replacement less expensive for the user.
The plug or shaft moves up and down within the polished barrel, offering a range of openings. Typically listed in bore sizes 0-4, 0-8, etc... the higher number represents the largest possible opening.