Interesting questions here. Firstly, there are a number of alternatives do Leased lines. They provide a similar service, but maybe are not as resolute as a leased line. These can include copper leased lines (SDSL) or even fully bonded solutions incorporating more than one circuit, to obtain a large amount of bandwidth.
Leased lines can be either fibre or copper. However if you are looking for more than 2Mb, then you will be looking at a fibre line. There are three main types. Either a 10Mb line, an 100Mb line or a 1Gb line. You would usually ask for this to be installed, then work out with your ISP (whether that be BT, NTL, Fluidata or ZEN) on the amount of bandwidth you are most likely to use. Eg. you may want a 100Mb line, but only want to use 10Mb of that, meaning that you only pay for the 10Mb.
However, they can be quite troublesome. As well as being extremely expensive if you are out of a city, they can take up to 60days to install and are frought with excess charges. However, if they work, they work VERY well.
For quick info, have a look at Fluidata Ltd | Leased Line | Legacy Internet Connectivity | Low Contention