Speech errors, often called “slips of the tongue,” are [usually] accidental deviations from the intended meaning during speech production. The study of speech errors is especially useful in investigating the process of speech production and the level at which syntax plays its role in the formation of sentences.

Three Basic Error Types
-Perseveration, when a phoneme from earlier in the utterance interferes with a word uttered later in the phrase (e.g. “I can’t cook worth a cam” for “I can’t cook worth a damn.”)
-Anticipatory, when a phoneme from later in the utterance interferes with an earlier word (e.g. “taddle tennis” instead of “paddle tennis”
-Exchange, when two phonemes are switched between two words, or two words switch places in a phrase (e.g. “queer old dean” instead of “dear old queen” or "when the paper hits the story")

The occurrence of these kinds of errors shows that there exists a level of representation where these phonological elements are represented in segments, and that there is a mental representation of phonological segments some time before they are produced (as illustrated by anticipatory and exchange errors). (1) The majority of these errors occur within clauses, which suggests that speech is organized into clauses prior to utterance. (1)

Additionally, word-exchange errors, such as “I left the briefcase in my cigar” for intended “I left the cigar in my briefcase,” show that words also exist as separate units in the mental representation. Exchanges never occur between function words (such as articles, demonstratives, etc.) and content words (nouns, verbs, etc), and word exchanges almost always occur between words of the same “parts of speech,” that is, nouns are switched with nouns and verbs with verbs, but rarely nouns with verbs. (1) This suggests that parts of speech do exist as aspects of our mental representations of lexical items. A counterexample might be raised by errors such as “we roasted a cook” instead of “we cooked the roast,” since in this context “roast” is a noun and “cook” is a verb, however, because both of these words can be nouns or verbs, this is not actually a contradiction.

Word exchange errors also illustrate that information about where in the sentence stress will occur is represented prior to retrieval of lexical items: in the clause “when the STORY hits the paper,” primary stress is placed on “story.” In the error “when the PAPER hits the story,” stress is retained in the place that it was originally intended, showing that stress is applied based on clause structure rather than being associated with a particular word. (1)

Syntactic constituents can be exchanged; in the error “there is an island on the small restaurant” for “there is a small restaurant on the island” an entire noun phrase “small restaurant” is moved. Movement of two words that are not part of the same constituent is never observed, that is, “there’s an island restaurant on the a small” would never be observed. Word exchanges always result in structurally well-formed sentences of English. (1)

Furthermore, speech errors demonstrate that the representation of bound morphemes occurs separately from the root lexical item they attach to, as illustrated by the error “you ordered up ending” for “you ended up ordering.” If the morphemes were attached to the roots at the time of the word exchange, the resulting error would have been “you ending up ordered.” (1)

Other applications
Signers of sign languages make many of the same "speech" errors as speakers (such as word exchanging), showing that signs and words are both stored as clusters of primary elements that can be recombined. Because hand movement is slower than that of speech articulators, signers catch and rectify their mistakes more often. Because these errors are easier to catch, study of sign errors may be useful in finding out whether speech errors occur because of word substitution, or whether (as has recently been proposed by phonetician Marianne Pouplier) phonetic speech errors occur due to a clash of motor commands as the speaker attempts to pronounce two different sounds at once. (2)

Speech errors are also used to study the language acquisition of children. Children only make speech errors with what they already know. Children acquiring language do not make syntactic errors (such as "sit down this immediately!", a blend of "sit down this minute!" and "sit down immediately!") until they have learned syntactic structures. This observation shows that children do not need to know syntactic structures when they begin combining words into two-word phrases around two years of age. (2)

(1) Cairns, Helen Smith. Psycholinguistics. PRO-ED Inc. Austin, Texas. 1999. Print.
(2) Erard, Michael. Read my Slips: Speech Errors Show How Language is Processed. Science, September 2007, 317(5845) accessed May 8, 2011: http://www.sciencemag.org/content/317/5845/1674.full

Further Reading:
Slips of the Tongue and Language Production, Anne Cutler, 1982
Errors in Linguistic Performance: Slips of the Tongue, Ear, Pen, and Hand, Victoria A. Fromkin, 1980