If you wish to visit a Schengen country for a holiday, to see the sights and experience the culture, you will need a short stay visa. This visa will allow you to visit any country in the 'Schengen area'. This visa can be given as a 'single' or 'mulitple' entry visa.
The two transit visas are usually only used when entering the Schengen country briefly to pass through into another country outside of the Schengen area.
The single entry visa will allow you to travel to the 'Schengen area' and visit any of the countries within it. However, you can only enter the 'Schengen area' once with this visa. When you leave the 'Schengen area' (for instance, by returning to the UK) you will not be able to travel to the 'Schengen area' again using that visa.
The multiple entry visa will allow you to travel to the 'Schengen area' and visit any of the countries within it. However with this visa you can leave the 'Schengen area' and then return to it using the same visa (if it has not expired). This visa is useful if you will travel within Europe a lot but need to leave the 'Schengen area' for reasons such as work or study.
You would only apply for this visa if you have a confirmed onward flight and so will be passing through the international transit zone in an airport.
This means that you are flying from one country to another, stopping for a few hours in an airport within the Schengen area. If you hold this visa, you are not permitted to leave the airport.
You would only apply for this visa if you will be travelling through the Schengen area on your way to a non-Schengen country. This visa is given for a maximum of 5 days and allows travel by coach, car or airports.
For example, you may wish to visit Belarus (a non-Schengen country) but travel there by car, passing through France, Germany and Poland. As you will travel through these countries, and provided it will not take more than 5 days, the transit visa would be applicable.