There is no standard way of notating chords. I even have a chart for a well-known piece my jazz ensemble plays that has different notations for the same chord! How is an inexperienced musician supposed to make sense of such things?
Fortunately, regardless of the style, most chord symbols tend to have the same basic form, and figuring out what a chord is supposed to mean is usually straightforward.
Root and Type
The simplest chord symbols represent triads (or inversions thereof).
The four chord symbols above represent the four types of triads - major, minor, diminished and augmented. Note that all four start with a note name - the 'root' of the chord. (If this means nothing to you, you might want to read this post.) The root is always a capital letter, and the largest single character in the chord symbol.
The C major symbol has only the root specified. The minor, diminished and augmented chords have an additional 'modifier' to indicate the triad type. This is a fairly consistent convention when naming chords - if no modifier is given to indicate that the triad is minor, diminished or augmented, then the chord has a major triad. Some musicians, however, use a modifier to indicate major chords, usually "ma" or "maj" (in lower or upper case)
The modifiers for the other triad types vary, depending on whoever is writing the chords. Minor chords can be indicated with a lower case 'm', as shown above, or with a minus sign (–), or "MI" (usually in small caps, or at least smaller than the root).
Diminished chords will usually be indicated with a small circle (as above) or with 'dim'. Sometimes the minus sign (–) is used. As noted above, the minus sign is also used for minor chords, so here we find a potential source of confusion.
Augmented chords will generally be indicated with a plus sign (as above) or with 'aug'.
Occasionally you may come across music where diminished chords are indicated as minor chords with flattened fifths, and augmented chords are written as major chords with raised fifths.
Extensions
Beyond the basic triads, we get extensions to 7ths, 9ths, 11ths and 13ths:
The image above shows how the extension to a major triad are constructed and the corresponding chord names. Here are the equivalent minor extensions:
This brings us to the next aspect of a chord symbol - the number indicating the extension. This number appears after the modifier for the triad - the number may be the same size as the root letter and on the same baseline, or it may be smaller and raised, or maybe just raised. There is no one way, and different musicians/publishers do their own thing.
One particular variation to this arrangement is the distinction between two types of 7th - the minor or dominant 7th, which is 10 semitones above the root, and the major 7th, which is 11 semitones above the root. The minor/dominant 7th is the most commonly used, so no particular symbol is used to indicate this, and the extension number is all that is needed. The major 7th, however, is always indicated in some way, the two most common being either 'maj' or a triangle (∆) placed before the extension number.
Sometimes, in the case of a major 7 chord, the 7 is left off, as the presence of 'maj' or ∆ is considered enough to imply that the 7th is present in the chord. (But note that earlier I said that some musicians use 'maj' to indicate simply a major triad - another opportunity for confusion.)
The diminished 7th chord has a slightly different structure:
The 7th in this case is based on the diminished scale, and the diminished 7th (interval) is 9 semitones, not 10. So Cdim7 (the name of the chord shown above) indicates a diminished triad with the diminished 7th added. You do not see further extensions to the diminished triad.
Alterations and Additions
Sometimes a note in a chord will be raised or lowered by a semitone. This is an alteration. The chord is notated as above, then the altered note is indicated after:
This is a 9th chord with a flattened or lowered 5th. As with extension numbers, the positioning of the alteration is not set, and may be on the same smaller, raised, etc.
Alterations may be indicated with sharps and flats, but also with pluses and minuses - yes, we've already seen those used for other things, so here is yet another instance where confusion can reign.
To make matters worse, on some music the pluses and minuses appear after the altered note rather than before it. (In fact, I have a piece at hand - a jazz standard - where the person who transcribed it has in some places used pluses and minuses before notes for alterations, in other places put the pluses and minuses behind the altered notes, and has also used the plus sign to indicated augmented chords! It's an absolute mess.)
Additions are notes added to a chord, but not as part of an extension. For example, we might add a 13th note to a major 7 chord, but as we are not added the 9th (or 11th) as well, this would not be a major 13 chord. So we write the name of the major 7 chord, then put the added note after:
Here we see two ways to indicate the added note - the first uses 'add', the second puts the added note in parentheses. Both ways are commonly used.
Sometimes the added note is not part of the normal scale for the chord, for example, a flattened 13th. If we add this note to our major 7 chord, we can simply append the added note without putting 'add' or parentheses, as there is no ambiguity about the extra note being an addition rather than an alteration:
A chord may have more than one alteration or addition:
Here there is both an addition (the flat 9) and an alteration (the flat 5). The two have been stacked in a column inside the parentheses. The same chord could have been written in a linear fashion: D7♭5♭9.
Omissions and Suspensions
Sometimes a specific note will be left out of a chord, e.g. we might omit the 5th from a chord with a flattened 13th:
The 'power' chord used in rock music consists of just root and 5th, so the common notation is, for example, Cno3. (The other common name for a power chord is a '5th' chord and the equivalent notation is, for example, C5, which is inconsistent with the way that extension numbers are typically used.)
Sometimes 'omit' will be used rather than 'no'.
A suspended chord will mean one of two things, depending on context. In pop and rock music (though it also crops up in other styles), we find 'sus4' and 'sus2' chords.
In these chords, the third is omitted and replaced with the 4th (e.g. Dsus4) or the 2nd (e.g. Asus2). The modifier 'sus' in these cases indicates a modification to the basic triad.
The other use of 'sus' comes from jazz, and is not (typically) followed by a number:
In this case, the meaning of 'sus' is to play a major triad one tone lower than the indicated root over that root. The 5th may or may not be played, and while the third is usually left out, occasionally it might be played as well. (Hey, it's jazz!) So E7sus would indicate a D triad played over a E, with possibly the B and/or G# included (or not).
Bass Notes
Sometimes a base note other than the chord root is used. The usual way to indicate this is to put the bass note after the chord symbol, separated by a slash:
Note, however, that this same format is also used for polychords (see next section).
Some Other Chords
Some commonly seen chords don't quite fit what has been described so far.
The 6th chord is very common. The meaning is clear - a triad (major or minor) with the 6th included. You could think of this as an extension or an addition, it really doesn't matter. D♭add 13 would have the same meaning, but the more compact form is the most common.
The 6/9 chord is also very common, especially in jazz, consisting of a triad with both 6th and 9th added. The usual notation is the 6 and 9 in a slanted fraction form, as shown above, but sometimes the 6 and 9 are stacked in a column, often (but not always) in parentheses. Sometimes '6add9' is written.
The 'half-diminished' chord may be indicated by a slashed circle, as shown. By 'half-diminished', what is meant is that the basic triad of this chord is a diminished one, but the 7th of this chord is a minor 7th (10 semitones above the root), not a diminished 7th (9 semitones above the root). This chord might appear with or without a 7 after the slashed circle, or might also be notated as min7♭5, which accurately describes the structure of the chord.
Polychords, also known as polytonal or stacked chords, are where one chord is played over another. A well known example from classical music is the 'Petrouchka' chord from Stravinksy's ballet of the same name:
This chord could be written as a Cadd♯6♯9♯11, but also thought of as an F♯ chord over a C chord, and written as:
or 
The first of these notations is exactly the same as the format for indicating bass notes. The second is clearer, but could take up more space vertically.
Occasionally on a score we come across problematic notations. For example, what is C4 supposed to represent? Does it indicate an added 4th, or a sus4 chord? Usually this has to be determined from the context. Occasionally it may be necessary to listen to a recording of the piece to identify what the chord is meant to be.
In a document I came across recently ("Standardized Chord Symbol Notation" by Brandt & Roemer, 1976), C4 was suggested as a notation for a type of chord used in jazz called a fourth chord. (You can read my explanation of fourth chords here.) But while fourth chords are fairly common in jazz, I have never seen such a notation used, and in fact such a notation misses the very point of what fourth chords are about.
Let's finish by examining a chord symbol (albeit a very unlikely one) and identifying the relevant parts:
The following diagram indicates the key points:
Using different styles for representing the triad type, seventh type and alterations, the same chord could be written as:
While the style might be different, the structure is essentially the same.
