In X10, "cut buffers" were introduced. These were limited buffers that stored arbitrary text and were used by most applications. However, they were inefficient and implementation of them varied, so selections were introduced. Cut buffers are long deprecated, and although some applications (such as xterm) may have legacy support for them, it is both not likely and not recommended that they be used.
The ICCCM (Inter-Client Communication Conventions Manual) standard defines three "selections": PRIMARY, SECONDARY, and CLIPBOARD. Despite the naming, all three are basically "clipboards". Rather than the old "cut buffers" system where arbitrary applications could modify data stored in the cut buffers, only one application may control or "own" a selection at one time. This prevents inconsistencies in the operation of the selections. However, in some cases, this can produce strange outcomes, such as a bidirectional shared clipboard with Windows (which uses a single-clipboard system) in a virtual machine.
Of the three selections, users should only be concerned with PRIMARY and CLIPBOARD. SECONDARY is only used inconsistently and was intended as an alternate to PRIMARY. Different applications may treat PRIMARY and CLIPBOARD differently; however, there is a degree of consensus that CLIPBOARD should be used for Windows-style clipboard operations, while PRIMARY should exist as a "quick" option, where text can be selected using the mouse or keyboard, then pasted using the middle mouse button (or some emulation of it). This can cause confusion and, in some cases, inconsistent or undesirable results from rogue applications.
There is a variety of clipboard managers available, and several desktop environments come with their own clipboard manager or have a clipboard manager intended for them (Glipper for GNOME and Clipman for Xfce); however, there are several DE-agnostic clipboard managers such as Parcellite (GTK+) and autocutsel (command-line), both of which are available in [community] and can be run as daemons.