A major triad (often represented by +) is composed of the root and the notes a major third and a perfect fifth above: hence C E G here.
A minor triad (-) has the root, minor third, and perfect fifth.
A diminished triad (o) has the root, minor third and diminished fifth. 
An augmented triad (x) has the root, major third and augmented fifth.

Given these rules, you should now know how to construct a triad in what is called root position. As illustrated above, the root note is in the lowest position, followed by the third and the fifth. 

To represent this chord, we can also use the figured bass system of chord symbols. Our example above begins on the first note of the scale of C major, therefore we use the Roman numeral I. In root position, the notes are a third and a fifth away from the lowest note, and this is represented too: . Usually, the 5 and 3 are assumed, but later you will place other numbers here.

There are other ways to write a C+ triad.

In first inversion, the third is the lowest note. The distance to the next note is a 3rd, and to the next note (the root, an octave above) a 6th. The chord symbol assumes the third and makes sure we know about the sixth: thus, . 
Second inversion places the fifth as the lowest note, making the distances between it and the other notes a 4th (to the C) and a 6th (to the E). The chord symbol is . 

Above is another set of triads in F minor. Can you tell how they have been constructed?
You may like to learn the shapes of these inversions and become able to identify them strictly by appearance!

Constructing and identifying triads

You will need to know about key signatures. Click here for a review.

When you construct a triad in root position, you can imagine the chord to be in the key of the bottom or base note. For instance, a root G would suggest G major or G minor. Let's say we want to create a major triad in A+ 

We'll begin with an A, since the key is A major. 

The next note is going to be C, and the next an E, so we'll put those down too, but wait! A to C is a minor third! The triad we have written is minor!

A C sharp would correct the situation. Note that we are in A major, which has some sharps in it. Because we are in that key (the key of the root, remember) we have to put the correct accidentals in.

We don't have a key signature, but we know that the sharps in A major are F, C, and G, so we'll put a C# in the correct place. Now we have a major third.

To write a given triad in 1st or 2nd inversion, write it in root position first, then move the notes (root and/or third) that you need to move. 
 

To identify a given triad, 
1) put it in root position if it isn't already.
2) imagine yourself in the key of the root note (which should now be in the bottom spot)

What are the sharps or flats in the key signature of that key? Do they occur in the triad? Check the intervals to see whether the triad is major or minor (see above.)

As a rule,
Major triads occur on the first, fourth, and fifth notes of major scales, and on the fifth and sixth notes of minor scales.
Minor triads occur on the second, fourth and sixth of major scales, and on the first and fourth of minor scales.

Download the triads worksheet » (Acrobat Reader required)

This name tells you everything. The V7 is always built on the "dominant" of a given key (the fifth note of the scale: for example in C+, the dominant is G.) and is made up of the major triad with another note a minor seventh above the root. Thus in A major: the dominant is E, and the V7 looks like this.

In minor keys we use the harmonic form of the scale, and have to raise the 7th note of the scale.
Consider a V7 in C minor, which has three flats (B, E, and A). B is the 7th note of the scale, so instead of an accidental B will be natural. 

There are three inversions of the dominant seventh. Again, in first inversion the third is on the bottom, and in the second inversion the fifth is on the bottom. In the third inversion, the seventh is on the bottom (and the complete triad is formed on top.)

Because the dominant seventh is built on the fifth note of the scale, we use the Roman numeral V in its chord symbols. The numbers again show intervals that help us to identify the inversion of the chord.

Again, to identify a V7, put it in root position (after figuring out what position it was in) to figure out the key.