Counterpoint/First Species

The first species of counterpoint is the most simple, being that of one note placed against one note (that is, if the cantus fimus is quarter notes, the counterpoint will also be in quarter notes).

There are two ultimate rules of counterpoint that should be observed in first species counterpoint and are almost never broken in any counterpoint with two voices.

When progressing into a perfect consonance, oblique motion or contrary motion may be used.
When progressing into a imperfect consonance, any motion might be used (however, parallel motion may only be used when progressing from imperfect to imperfect).

The Definition of Motions

Motion, with regards to counterpoint, is the relationship between two notes in different voices and how they "move", with reference to each other, as the score progresses.

Motion where one voice does not move and the other does is called oblique motion.
Motion where one voice moves up and the other down, or vice versa is called contrary motion.
Motion where both voices move up or down in the same interval is called parallel motion.
Motion where both voices move down, or both up (but not necessarily by the same interval) is called similar motion.

Parallel motion may be considered a special case of similar motion.

The Definition of Consonances

A consonance is any interval considered harmonious by tonal standards (the third, sixth, fifth, octave, and unison). (Note that the unison may not be used except in beginning a counterpoint below the cantus firmus.)
- a Perfect Consonance is any consonant interval that cannot be major or minor. These are the Perfect Fifth, Perfect Octave, and Perfect Unison (and any variation of these such as the fifteenth).

Special Note: In traditional counterpoint, the fourth is not considered a consonance because of the potential dissonance that can result if both fourth and third or fourth and fifth are utilized with a bass line in three or four voice counterpoint; however, in the less strict counterpoint of more vertical music, fourths are very frequently utilized as consonances. In the case of any exercises in counterpoint, it would be prudent to consider the fourth as a dissonance.

an Imperfect Consonance is any consonant interval that can be either major or minor. These are the major and minor third, sixth and tenth.

A dissonance is any interval that is considered to be unpleasant to the ear or unstable by tonal standards (these are the second, tritone, seventh, and augmented or diminished intervals)

Linear Nature of Counterpoint

Good counterpoint must be multiple melodies set against each other. As such, both melodies must be well written, with mostly stepwise motion and a high or low point that is reached as a "goal" of the melody. In general, large skips should not be made over the bar line or onto a strong beat. There are many other rules that will not be described in particular detail here.

Chordal Nature of Counterpoint

It is important for counterpoint to make harmonic sense in addition to melodic sense. The first and last chords must be the tonic chord of the key, and it is important for the cadence to make sense (usually the dominant or subdominant chord of the key).

Archaic vs. Modern

Fifths are more utilized in archaic style as a perfect consonance. In modern style, fifths are less utilized in favor of imperfect consonances, because these emphasize the tonality of the implied chord (major or minor)

Use of Leaps

According to Fux, leaps of seventh or any diminished or augmented intervals should be avoided. The melodic direction should change after a leap greater than a third.

Cadences

'in archaic style': If the counterpoint is in the upper voice, the cadence should involve a major sixth resolving to a perfect consonance. If the counterpoint is in the lower voice, the cadence should be a minor third resolving to a perfect consonance.

'in modern style': A V or IV chord must resolve to a I chord.

References

Chapter 8 of Music In Theory and Practice by Benward and Saker