Actually, a period is a higher level of organization than a phrase, not a special kind of phrase.
Levels are: Section, period, phrase, semi-phrase, motif. In some pieces, section and period might merge, or phrases can't be divided clearly into semi-phrases...
Types of phrase: Mostly two, defined by the cadences (inconclusive phrases, acting as questions, and conclusive ones, or answers). Cadences are what you need in order to write phrases in a tonality.
Balance is a matter of the particular style. A classical period would be made up of two phrases, highly symmetrical (say, four bars). First phrase, inconclusive, with slow or static harmonic motion (few chord changes), and second phrase conclusive with active harmonic rhythm (lots of chord changes). Baroque and Romantic phrases and periods can have their own particularities. Of course, phrases do not need to be 4 bars long, and periods do not need to have two equal lenght, symmetrical phrases; also they can have more phrases than just two.
As per rhythm, question and answer might share similar motives (both rhythmic and melodic); the most extreme case would be when they are identical except for the ending, or they can be contrasting.
A link with some explanations examples. Very basic stuff, but it might help:
http://www.teoria.com/tutorials/forms/phrases-periods.php