Modification of the form or sound of a word or morpheme under the influence of an adjacent word.
(A morpheme is the smallest grammatical unit in a language: it has meaning, but no smaller part of it has meaning. For example, 'unforgiveable' has three morphemes: 'un' and 'able' are bound morphemes, and 'forgive' is a free morpheme.)
One example of sandhi is the use of 'a' before words starting with a consonant (a hawk) and 'an' before words starting with a vowel (an owl).
Another is the phrase "cats and dogs", where 'cats' ends with an 's' sound but 'dogs' ends with a 'z' sound.
Those are both examples of external sandhi: changes at the boundaries of words.
Internal sandhi is a change within a word; the Wikipedia example is 'sympathy' (syn- + pathy).
Etymology: 1800s, from Sanskrit samdhi, 'placing together'. This sort of word modification is more common in Indian languages, and especially Sanskrit, hence the name.