A progression is simply a movement of a planet (or sensitive point such as the Ascendant). The Sun progresses about one degree per day, and thus completes a circle of the Zodiac (and your horoscope by transit) in twelve months.
This is the actual measurable movement as laid out in an ephemeris. However there are systems that use this movement in a symbolic way rather than a literal way.
The most commonly used alternative to transits is the system of Secondary Progressions. Instead of the one degree daily movement of the Sun being taken literally, each day is taken symbolically as a year in your life, starting with the planetary positions at your birth, the planetary positions for twenty four hours later is taken symbolically as a chart for you aged one - this can continue through your life - so the 50th complete day after your birth represents you at age 50. The Progressions are read much in the same way as Transits but the Moon becomes much more important. This is because it is the fastest moving planet and as progressions slow everything down by a factor of 365, any changes of Aspects or changes of House by progressed planets are nearly always those made by the Moon. After the Moon, Mercury Venus and the Sun are the next fastest planets and they may change House or Aspect in a year (though it will take the Sun on average 15 years to change signs after your birth and then it will change signs every 30 years or so. The progressed Moon will change signs approximately every year and you will experience the Progressed Moon returning to its starting point in your chart around the age of 29 - 30 - the same time as your first Saturn Return by Transit. Both are taken as important symbols of adulthood and the main stages of our life.
There is usually little point in progressing Uranus, Neptune and Pluto as their progressed position may change little in the equivalent of 70 or 80 years - they may not form any new progressed aspects at all or change house or sign.
There are other progression systems, based on a day for a lunar month or a lunar month for a year. However these are not frequently used compared to Secondary Progressions.
Many Astrologers who specialise in forecasting will look at Progressions in association with Transits. A common view is that Transits will 'release' the energy of Progressions, for example if the Progressed Moon aspects your natal Venus and this in turn is being transited by Mercury, the Mercury transit will activate the Progressed Moon to Venus aspect. If the aspect is an easy one then your popularity, social life, relationships, and appearance will all blossom. You will express yourself in a more harmonious and beautiful way. The point at which the Mercury transit forms an exact aspect to natal Venus or your progressed Moon is the time at which the progression begins to effect you.
There are Astrologers who will argue that Progressions have an internal effect and Transits an external effect so a Transit by Mars will manifest in a different way than a Mars Progression. However this view seems to be losing ground.
Astrologers who specialise in forecasting will probably use the Progressed Moon as a major part of their system, together with the Transits of Jupiter, Saturn, Uranus, Neptune and Pluto. The Progressed Sun and Progressed Mercury, Venus and Mars may be used in preference to Transits by these planets (whose effects may only last a few days at most).
One of the possible advantages of using progressions is that your entire life is laid out in the first seventy or eighty days after your birth and it can be interpreted using the normal rules of astrology (though with much smaller orbs).
I could say a lot more but I hope the above gives you a general idea.
