I've recently started learning and composing music. I was writing a simple melody and chord progression in 4/4 in some music software. It was just a basic 4 bar melody with 4 different chords one on each bar. I decided for fun to delay my chord progression by a 8th note and it actually made it sound better. It sounded a bit more jazzy/swingy. I think this is to do with beats or something and playing off beat but I'm still learning and don't really know what I did.
An anticipation note has its classical root, for example most notably in the cadence. Note are played slightly ahead ("out of phase") to create liveliness. I think this makes the division of rhythm finer, from (say) quarter to semiquaver, becoming essentially a dotted note.
A similar technique is also common in musical performance, by which actor act slightly ahead of the singing of that corresponding note.
In jazz, anticipation is common, contrary to classical tradition. The instances are numerous, for example the popular song "Let it Go". Or the popular TTB piece Praise his Holy Name is also rich in anticipation note.
Personally I think a long series of uniformly anticipated notes create fatigue. Anticipation note shall be used sparingly, to preserve its unpredicted nature.
Aminopterin is correct: It's some form of melodic decoration. It's not an anticipation note, it's some form of suspension, I believe.
The reason it is a bit unusual in this specific circumstance is because usually it refers to one note of a chord being written after the rest of the chord, but you appear to be talking about the whole chord written after a note from it (from it, hopefully...;-P)
A reason why I think it's a suspension as opposed to an anticipation is because an anticipation, by definition, starts before the beat where the actual chord is supposed to start. What you have sounds to me to be a note, from the chord or the melody, played on the beat, followed half a beat later with its chord. This sounds like suspension to me.
My basis for this answer is from a little webpage from a little website that I came across about a week ago, and thoroughly enjoyed. It's worth a read:
Delaying the chords by an eighth note means that you are playing on the "off" beat or upbeat. This emphasis on the upbeat can be described as syncopation. Syncopated rhythms are found in all sorts of music depending on how it is played. It is a big part of Jazz as you mentioned. Also, pop, reggae, ska, polka, come to mind. It can be incorporated into and can be found in most any type of music.
Delaying or advancing a chord change by a bit is a common and very satisfying way to play music. It adds a lot more 'romance' to the music when done in the right situation. I'm really impressed that you found this out on your own.
People love anticipation and surprise. It makes the music interesting. Try not to overdo it. In some songs, though, you can overdo it as much as you want. It forces the listener to think a little more but, in my experience, the listener is more than happy to oblige.