TH55 - remember the difference between bits and bytes too.
If you are paying for a, let's say, 30Mbps connection from your ISP (30 megabits per seconds), that translates into a theoretical perfect download speed of 3.75MB/s (megabytes per second). Add in congestion in your local area from all the other folks using the net, a little bit of interference etc and you will usually get about 75% or so of that.
When you then start transferring files over wireless, there is an overhead there too and it can sometimes be a big one, especially with old 802.11g routers that are short range. You are using a very congested part of the radio spectrum and all your neighbours' routers, microwaves and cordless phones or baby monitors are using it too. Not to mention every bluetooth device in the area
Your G router will have a limited range and power. Off the top of my head the effective range will be somewhere around 10-15m or so. The improved 802.11n routers will have two or three times that. This range is dramatically affected by intervening walls as well as other wireless interference.
I would first contact your ISP and say you believe you are not getting the throughput that you are paying for and believe you need your line checked by an engineer. They will check the power levels and signal on your line but first they will ask you what download speeds you get off-peak on a wired connection. They mean an ethernet connection straight to the router itself, not wireless. If you get anything above about half of what you are paying for, they will point to the tiny "up to" underneath the advertised speed and you are on your own.
As for Browns2212's neighbour with the strong signals, there are two causes of that. One is that they may have boosted their transmitting power and each country have laws about the power available on the 2.4GHz and 5GHz bands. The second reason is that the neighbour may have fitted a high gain or even directional antenna.
All antennas typically found on a wireless router have a transmission pattern resembling a flattened donut (ignore the directional ones for the moment). A low-gain omni-directional antenna, say 2dB, will transmit in a nice, fat donut but the power won't go very far. The higher the gain on the antenna, the further the signal goes but the flatter the donut gets. A high-gain antenna of, say 9dB will transmit the same amount of energy as your fat donut 2dB one, but only in a flat ring around the antenna. Above and below there will be little to no signal.
It is possible your neighbour has a high-gain antenna serving just a single floor of their building with no reception (from that antenna) on the other floors. Of course, if they have a directional antenna to send wireless down to the end of the garden to the shed, you could just be in line of sight of that beam.
So, the first thing to check is that on a wired ethernet connection you are getting the speed you are paying for. Remember the difference between the advertised speed and the real download speed (a factor of 8).
Once you have confirmed that you are getting what you are paying for, you may want to look at an 802.11n router. Anything generic with a couple of external antennas would do. Internal antennas are fine if you are not too worried about the range and there are not too many other wireless users near you. Remember to check that the router's firmware has been updated to the latest version from the router provider (don't worry about third party firmwares from the likes of Tomato, Toastman, Shibby or Merlin for now - they are for users who either have specific wireless networking needs or want to have complete control over how their router behaves).
If you have any newer wireless devices that use the 5GHz wireless band (such as newer smartphones or tablets) then a dual-band router would be good. The 5GHz band is generally less congested than the 2.4GHz band (though slightly shorter range). You can pick up a Cisco E3000 for not too much these days or if you want to splash out, the ASUS RT-N66U is well respected but more expensive.
Any more questions, just ask us.