Samples are most commonly used on a sampler, which is a piece of hardware configured by the user with song samples. Computer programs are also used. Samples consist of one part of a song. There are many variations of a sample, such as: beats, instrumentals, a capellas, breaks, loops and spoken word (film/ tv lines).

Early music sampling
One of the earliest samples was used in 1961 by James Tenney in his song Collage #1 (“blue suede”), which used samples from Elvis Presley’s “Blue Suede Shoes”. The Beatles also used the technique on a number of popular recordings, such as “Yellow Submarine” and “I am the Walrus”.
DJ sampling was reportedly initiated by DJ Kool Herc in the early 70s but did not truly take off until the early 80s, when hip hop producers Grandmaster Flash started making rap albums using sampled breaks instead of live studio bands. The creation of rap catapulted music sampling to the forefront, largely for the purpose of extending the breaks for b-boys (break dancers) to put on exhibitions at party’s and clubs.

Legal issues
Sampling, from a legal standpoint, has been greatly contended since the popularity rise of rap music. Originally, asking to use a sample wasn’t (didn’t seem) necessary, but as soon as songs that used samples started making money the original artists began claiming copyright infringement. Most sampling cases are settled out of court and usually conclude with: paying for the royalties, handing over copyright of the song (to the original artist), cancellation of the song release, or replacing the sampled portion of the song.

Sampling Lawsuits [1]
Grand Upright Music Ltd. v. Warner Bros. Records, Inc.
Campbell v. Acuff-Rose Music
Tuff ‘N’ Rumble Management, Inc. v. Profile Records
Newton v. Diamond

Some songs that use samples [2]
“Bittersweet Symphony” – The Verve
“Ice Ice Baby” – Vanilla Ice
“The Adventures of Grandmaster Flash on the Wheels of Steel” – Grand Master Flash
“Pass the Mic” – Beastie Boys

Modern Sample usage
The most belligerent use of music sampling can be seen in the musical technique called mash-up. Mash-up, usually created on a computer, a song that consists of two or more songs, usually by overlaying the vocals from one onto the beat of another. The most prominent mash-up artist at the time is Girl Talk (Gregg Michael Gillis), who uses up to 20 different samples for each song. Gillis cites fair use doctrine under the Copyright law of the United States for his ability to use samples. Here is an example
Girl Talk – Bounce That