An anticipation is an unaccented non-chord note which is an early sounding of the following chord note, which is then repeated within its own chord.
For example, the soprano chord notes B-C here can be decorated with a C anticipation. The C initially sounds before the C major chord, then it is repeated with the C major chord.
Anticipations are normally fast notes such as quavers or semiquavers (8ths or 16ths).
Anticipations are normally only found in the soprano part, although they can also be used in an inner part if they are combined with a soprano anticipation. They are not normally found in the bass.
In Chorale style (hymns set in SATB 4-part harmony), anticipations tend to be found only at cadences, and only relatively rarely. In other types of melody (whether vocal or instrumental), anticipations are quite common.
The song Home Sweet Home has a few examples of anticipations, which are used along with other types of melodic decoration to create a unified rhythm across the melody:
(a=anticipation, x=auxiliary note, p=passing note).