FIGURED BASS
Figured bass notation, also known as "chord symbols", always includes, using Arabic numerals, the intervals above the bass note where a note is to be played. When 5 and 3 are present (i.e., in a root position triad where the intervals above the root are a 3rd and a 5th) they are usually omitted unless shifting back to root position from an inversion while keeping the same notes. 

In harmony, the notation also shows the degree of the scale (I through VII) of the given key on which the chord is built. The examples below show some examples. In the top 2 staves, the triads are built on the tonic (I) of the key. The bottom stave shows dominant (V) 7ths, which are built on the fifth degree.

Just for fun, here are the degrees of the scale:
I - tonic; II - supertonic; III - mediant; IV - subdominant; V - dominant; VI - submediant; VII - leading note.
 

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.

The perfect cadence V-I
- What key are you in? Figure out what your dominant and tonic (V and I) will be. Put them in the bass. In our example, we have D and G.

- Write out the note names from the dominant triad and the tonic triad (here, DFA GBD). Remember, we have four voices to deal with, so one note (the root) of each triad will appear in the bass. The triads themselves will appear in the top three voices.

- Choose a note from the dominant triad to be in the top voice (the 'melody'), and write it in the top stave. I selected A. 
- Write the rest of the triad below it (in close position, meaning in the nearest possible line or space corresponding to the right note name) making sure you don't go below your bass note.
- Which note from the dominant triad also appears in the tonic triad? (here, it is D). Write this note in the same voice in the next chord (here, in the next bar). Your progress so far is illustrated below.

- Now, write the rest of the tonic triad (here, G and B) moving as smoothly as possible. You must use these rules:
--- The common note to both triads appears in the same voice, of course.
--- Move the other two voices 'up' one step: here, the F goes to G and the A goes to B.

Symbolize the chords:    V      -        I

A memory aid for resolving the perfect cadence is "2 up, 1 common".
 

The plagal cadence IV-I
Follow the same procedure, more or less, working out your triads and a common note. Put the subdominant and tonic in the bass. There will be a common note again, BUT
- this time, you will be moving the other two notes of the triad DOWN one step

Symbolize the chords:     IV     -      I

A memory aid for the plagal cadence is "2 fall, 1 common".
 
 

The imperfect cadence I - V
Follow the procedure for the perfect cadence, but reverse it. In other words, write the tonic going to the dominant in the bass, and move the tonic triad to the dominant triad in the top 3 voices, making sure 2 fall, 1 note is common (i.e., reverse of the perfect cadence). 

                                       I       -      V
 

The imperfect cadence IV-V
Write the subdominant going to the dominant (up a 2nd) in the bass. Work out the subdominant and dominant triads. There will be no common note. All notes will move down, in close position, to the notes of the dominant triad. Two will fall a second, one will fall a third. Thus:

                                      IV      -     V

A memory aid for the IV-V: "2 fall a second"

The principal degrees are I, IV, and V. These form the basis of much harmonic structure, usually with repetition of the tonic to establish the key. It is in the nature of our harmonic language for a V chord to want to go to I.  IV is a good intermediate step, and the fact that IV and V share no common notes gives some variety. 

In order to move through a progression such as I - IV - V - I, you can use the rules given on this page for proper voice leading (a clever name for "where do you put the notes"). 

- Reverse the plagal cadence to get I - IV. IV to V we've also seen before in an imperfect cadence, and we also know V - I. 

                         I       IV     V          I

Note how the top voices move as smoothly as possible, keeping all common notes in the same voice, and moving by step except between 2 notes from IV-V (a third, as described above.)

If you are given a V7 instead of a V, put the third, fifth, and seventh in the top three voices and move them as smoothly as possible. If you make the third and seventh of the V7 fall a second, this should be possible.