Standing committees are part of the process by which a Bill becomes part of statute law in the United Kingdom. After the two readings in the House of Commons, but before a final vote, the Bill is passed to a standing committee. It is here that, in theory, scrutiny and amendment of the Bill takes place.
Unlike select committees, standing committees are set up as and when they are needed. They operate along party political lines, and political parties are represented in proportion to their number of seats in the House of Commons.
As mentioned above, it is the purpose of standing committees to scrutinise legislation. This is where legislation is considered in depth (its first reading in the House of Commons being little more than a summary) and amendments are made to parts which are unnecessary, unethical (for want of a better term), unenforceable or contradicting other laws. In theory, the standing committee stage is the most important of the legislative process, as this is where backbench (non-Cabinet) and opposition MPs have some say in the shape of the final Bill. It is also where any problems should be solved. Scrutiny of existing and proposed legislation is considered to be one of the main functions of Parliament and, along with the debate, the standing committees are where this takes place.
In practice, however, effective scrutiny does not always happen. The main reasons for this are the party political nature of the committee, and the government's imposition of the guillotine. What often happens is that the Opposition1 object to most of the Bill (which has, as a rule, been proposed by the government; there are very few Private Members' Bills) on principle, and immediately start proposing all sorts of amendments. This takes time which could be used for proper, non-party-based scrutiny. Eventually the government becomes impatient with the committee and imposes the guillotine, which is a time limit on the committee's deliberations. In this way, the Bill reaches its third reading in the Commons without being properly checked, and bad legislation gets passed.
If you want to know more about how Bills become Acts, I recommend webtoe's excellent writeup under House of Commons.
1 Opposition with an upper-case 'o' refers to the non-government party in a two party system. For instance, in Britain, if Labour are in power, the Conservatives are the Opposition, and vice versa.