Germanic umlaut is a sound change that affected all the Northwest Germanic languages (i.e. all except the members of the extinct Eastern branch). It affected the back vowels /a o u/ in a syllable where the following syllable included /i/ or /j/, causing them to front, generally, to /ɛ ø y/. At first this change was allophonic, but if the later syllable with /i/ or /j/ was dropped, then the change became phonemic, as the only difference between the two words was the umlauted vowel.
In English, umlaut is the reason for many of our irregular plurals, like foot/feet or mouse/mice (these pairs being fōt/fēt and mūs/mȳs in Old English). The Great Vowel Shift goes a long way to explaining why our umlauted vowels (and their non-umlauted counterparts) don’t match the pronunciations of their equivalents in other Germanic languages. In a historical phase of the language it was also common to derive a transitive verb by adding a suffix containing /i/ to an intransitive verb, so we have some word doublets descending from that time, like to sit/to set or to fall/to fell. Umlaut is similar to, but not the same as, the phenomenon of ablaut that is used to conjugate our Germanic weak verbs.
In English, it’s not visibly obvious in the written language which vowels are the result of umlaut and which aren’t, but in some other languages it is much clearer. German, for example, uses the letters ⟨ä ö ü⟩ to denote the umlauted vowels, and Swedish uses ⟨ä ö y⟩. (With some exceptions; sometimes they don’t denote a word that has undergone umlaut, but a borrowed word that happens to use the same sound as a native umlauted vowel. And then there are some other exceptions where umlauted and non-umlauted words diverged in meaning, or where other sound changes have made it desirable to use a different vowel.)