Download from here a very useful leaflet from the Electoral Reform Society illustrating how STV elections are counted.
If you want to count an STV election by hand (say if you are running a demonstration election from a street stall, or as part of the a public meeting), you need to browse and print the detailed rules recommended by the Electoral Reform Society from here. Alternatively you can buy a printed copy from the Electoral Reform Society.
Alternatively you may feel that the leaflet gives you sufficient idea of the counting rules and wish to let the computer do the work. We can recommend Open STV http://www.openstv.org/. Annual licence for an individual or organisation with 0 to 3 employees is 5 USD. The counting program is run offline, but you can use the associated software Opavote to capture votes online. This is free for small elections.
So far most public elections using STV have been hand counted. However when STV was introduced for local elections in Scotland in 2007 it was decided to use machine readable paper ballot papers and to count them by computer. By using paper ballot papers the concerns that have arisen in the USA about voting machines have been dealt with; there is an audit trail.
Significant sized organisations will typically use an independent supplier of ballot and election services, such as Electoral Reform Services.