Ornaments (i.e. decorations of the melody) have always been very common in music. These days, R&B singers in particular - especially when singing ballads - use ornamentation to an extreme degree. (They even do it when singing the American National Anthem, and no-one bats an eyelid, although it always sounds a bit odd to me.) But this has been the case for hundreds of years - in renaissance and baroque times, in particular, a solo performer wouldn't be expected to play or sing just the notes on the page. They would make up their own ornaments. (Often, in a piece with repeats or a D.C., they would perform the melody more or less unembellished the first time, and then add the ornaments on the repeat.)
Sometimes, the composer would specify exactly what ornaments he wanted, rather than leaving it to the performer (you can see that this makes sense if there are several people performing together!), and symbols evolved for various types of ornament. There were (and are) different ways of interpreting these symbols, varying according to when and where the music was written. In most cases, performers today do not know (for sure) how ANY given ornament should be interpreted, but there are some general guidelines which are usually applied in the first instance, until the conductor specifies something different.
(I'm only going to talk about ornaments used in choral music, but the Wikipedia article has the others, plus fuller explanations of the ones I WILL mention.)
There are two main types of ornaments commonly used in choral music: trills (including mordents), and grace notes (including appoggiaturas and acciaccaturas).
Trill: everyone knows what a trill is, I think. The symbol is tr placed above the note, and this indicates that the performer should alternate rapidly between that note and the one above. (How do you know whether a trill on an A goes to B or B flat? It depends on the key signature. If there is a B flat in the key signature, then a trill on an A goes to B flat; otherwise, it goes to a B natural. If the composer wants to specify a trill to a note that's NOT in the key signature, the trill symbol has an accidental above it.) The trill itself is straightforward enough to interpret; the difficulty is with the start and end of the trill. In general, trills in music written before about 1830 actually start on the note above, and trills in more recent music start on the note itself. The END of the trill usually has a little twiddle involving the note below. (There are exceptions to both rules, but the conductor will specify these.)
The examples below should give you the idea. Note that the number of repetitions of each note will depend on the speed and style of the music - there is not a hard and fast rule that determines how many there should be. Also, the fact that I've put a triplet in the last example does not mean that that's how it would be performed - the notes in the trill are not necessarily all exactly the same length, and I only put the triplet in so I could include all the notes.
Mordent: this is the "squiggle" symbol. Sometimes (particularly in earlier music) this symbol implies a trill, in which case it would be performed as above; however, usually it implies just a single twiddle to the note above, as shown below.
Grace note: these have two general characteristics, plus some more specific ones that I'll mention under each type. The first general characteristic is that they're printed as small-size notes (i.e. smaller than the "normal" notes). The second is that their length is not counted towards the length of the bar. There can be several grace notes, or just one. If there's only one, it will be an appoggiatura or acciaccatura - more on each in a minute. If there's more than one, they're usually just called grace notes.
Mozart's "Ronda Alla Turca" (the last movement of his piano sonata no. 11 in A major, K331) has some good examples of grace notes. Look at the first page. There are grace notes in the right hand on the second line, and in the left hand on the fifth line. (There are instructions at the bottom of the page regarding how to perform these, but they're likely to be written by the editor rather than Mozart himself, I suspect. Also, I've usually heard this piece performed NOT according to those instructions!) In the first bar of the second line, you'll notice that if you ignore the grace notes, the four full-size quavers add up to two full crotchet beats, i.e. a full bar of 2/4. That's what I mean by the length of the grace notes not being counted towards the length of the bar.
I've usually heard the aforementioned grace notes performed like this (I've omitted staccato marks etc. for clarity):
but the editor in this case is suggesting that the correct version is this:
In general, I would say that the first method of interpreting multiple grace notes (i.e. before the beat) is more common, but the conductor may specify the second.
Acciaccatura: this should really be pronounced "atch-ACK-a-toora" but is usually closer to "ack-see-ack-a-toora" or something similar. It's most often translated as "crushed note", and it's the single-grace-note-with-a-line-through-it symbol. It's played as close as possible to the main note. Usually, if a choir has to sing one, they will be instructed to sing the acciaccatura on the actual beat and then move immediately to the main note. This is because if a choir tries to do it the other possible way - i.e. to sing the acciaccatura at the last possible moment BEFORE the main note - the result is usually rather untidy! However, when played on instruments, it's more common to play the acciaccatura before the beat. For example, the intro to The Shepherd's Farewell:
Appoggiatura: pronounced "ap-PODGE-a-toora", and most people can manage that (although you do often hear "a-podge-a-tyoo-ra"). This is much more common in choral music than the acciaccatura. It's the single-grace-note-WITHOUT-a-line-through-it symbol, and it's usually translated as "leaning note". Unlike the acciaccatura, which is always as short as possible, the appoggiatura has a specific length, although (like other grace notes) the length of the appoggiatura does not count towards the length of the bar. Also unlike the acciaccatura (which is usually written as a quaver, however long it actually is), the appoggiatura can be written as more or less any note value, but the written note value doesn't necessarily imply how long the appoggiatura should be! This ornament causes more confusion than all the others combined, I think.
As a general rule, the appoggiatura note is performed on the beat, and the main note comes later (even if the appoggiatura is written before the barline). In most cases the two notes (appoggiatura and main note) are of equal length, UNLESS the main note is dotted, in which case the appoggiatura note is twice as long as the main note (i.e. if it's a dotted crotchet, the appoggiatura will be a crotchet and the main note will be a quaver). As usual, there are exceptions to these rules, and the conductor will specify them. But, to give you an idea of the common practice:
Those two examples are both from "He Was Despised" (Messiah), and (like many things in Messiah) will often be interpreted differently. However, they will hopefully clarify the most common way of performing appoggiaturas if you are given no instruction to the contrary.
I could write much more, but I think I've covered the main bits I intended. (Note to self: stop coming up with ideas that take much long than you expect!)
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