London City is the closest, it is in Zone 4 on the London Underground map as opposed to Zone 6 for Heathrow.
wrixonshea whereabouts in London are you visiting? The options are as follows:
Heathrow (LHR): 15 minutes to Paddington (London's West End) by the extremely expensive Heathrow Express, 30 minutes to Paddington by Heathrow Connect, 50 minutes to Leicester Square. Heathrow is good if you are heading to somewhere in West London or anywhere along the Picadilly line. Airport charges can be very high and congestion on the airfield can be quite considerable at busy times. Airport can be time-consuming to navigate if you are unfamiliar and depending on the terminal you fly through it can take a bit of time to get to the Tube.
Gatwick (LGW): 30 minutes by the (overpriced) Gatwick Express to Victoria, 35 minutes by the (much cheaper) Southern to Victoria. 35 minutes to London Bridge by First Capital Connect, 40 minutes to City Thameslink by First Capital Connect. Gatwick suits West and South London (though it has an easy access to North London too as the First Capital Connect passes right through the city). The airport is halfway to Brighton but it has good rail access into several main London stations.
City (LCY): 22 minutes by DLR to Bank station. Simplest one to arrive into but it works best if you are in East London, Docklands or the City. Flights can be pricey though! Majority business passengers. DLR is right outside the front door so it is a very simple transfer to public transport
Southend (SEN): 50 minutes by Great Anglia train into London Liverpool Street. Again suits City and the East End. It can be a very cheap option sometimes. If you are considering Stansted or City then I would always look at Southend too. Very small airport and very quick to process through, train station is right outside.
Stansted (STN): 45 minutes (supposedly) by Stansted Express to London Liverpool Street. Again suits City and the East End. I don't like it as much as Southend though, it is a bit of a trek to the train and to the plane! Can be a pain to travel through as it is a Ryanair hub so it can be a tad chaotic if you are travelling at the wrong time
Luton (LTN): 10 minutes by bus to Luton Airport Parkway, 35 minutes to St Pancras by First Capital Connect (just to note, Luton and Gatwick are on the same trains, FCC travels from Brighton to Bedford, takes in both airports and a swathe of city centre stops). Bit of pain having to get a bus to the train station. Best airport for North London though.
Basically when I am flying into London I decide based on where in London I will be going to. While the city has an excellent transport network it can be a pain to navigate with luggage and especially in summer you don't want to be stuck on the Underground for too long! Use
www.tfl.gov.uk to work out your transport links and decide from there!