Why are some stations "left-handed"?

More specifically

White City is left handed because the tracks are back to front (because of the original configuration of the depot)
Notting Hill Gate (I think) Chancery Lane and St Pauls (definitely) were, as trt said, built with the tracks at different levels because of width constraints - there was simply no room to have the statoin tunnels side by side, never mind far enough apart to squeeze a lift or escalator shaft in between them, so both platforms are accessed from the same side. Westminster (Jubilee) is arranged like this too. (Interestingly, the eastbound at St Pauls is above the westbound, but the converse is true at Chancery Lane)

Holborn is the oddity. It was added later than the rest of the line, to provide interchange with the Picadilly (replacing British Museum). As the running tunnels were too close together at that point to put a platform in between, separate side platforms had to be built.

The Central Line is actually more consistent than most - the Victoria Line weaves over and around itself and various other lines so much to provide cross platform interchanges, that it is very hard to remember which side any particular platform is.

Tufnell Park is an interesting one - it is in the middle leg of an S-bend. To ease the curvature the tracks cross over each on both curves, so that the running lines are back to front as they pass through the station.

The Northern Line through the City (London Bridge/Bank/Moorgate) was built with left hand running so that the first tunnel to be built, which is the eastern, one would be used by the northbound trains. The second tunnel was built at deeper level, and it was expected that the low-powered locomotives would be incapable of making the climb from that lower level into the King William Street terminus, especially as it was at the extreme limit of the power supply (the power station was at the far terminus). In fact, in many cases they proved incapable of managing even the shallower gradient.

For the early lines the companies had to buy the land above the tunnels.

Chancery Lane and St Paul's have the platforms above each other to minimise the cost.

The westbound Piccadilly platform @ South Kensington is also left-handed, a throwback to the original construction of the station to provide interchange with the erstwhile deep-level District plans.