Shimano are the dominant manufacturer of gears, transmission and brakes for bikes (others include Campagnola, SRAM, and various smaller outfits).
While many bikes bought 'off the shelf' contain a mix of products from different ranges, there is a clear pecking order for the Shimano components that you will need to be aware of if comparing bikes.
Be aware that just because the cheaper components are, well, cheaper, it doesn't mean they are no good. Just less good than the better ones. The gears are still going to change, and the brakes are still going to stop the bike...just with more elegance and rigidity, and less weight, towards the upper ranges. The more expensive ranges do also tend to look nicer.
This is the entry range for Shimano, and commonly supplied on budget bikes. Functional rather than glamorous. The main disadvantage is with the gear shifter levers, where two separate levers are used for shifting up or down gears (all the other ranges have just one lever, used in conjunction with the brake lever to change gears. This is most important when you have your hands in the 'drop' part of the handlebars, when the second gear changer lever is rather inconveniently placed.
The Shimano Tiagra range is still used largely on budget bikes, and represents a small step up from Sora. Perfectly competent and will work long and hard without causing any problems (sample chainset price* £70)
The most common system found on mid-price bikes, many bikes are supplied with some Shimano 105 parts, others with the complete range. This is the most popular range - light, efficient, and reasonably priced - with the higher ranges you pay quite a price premium for smaller increases in performance. The majority of leisure cyclists will be perfectly happy with Shimano 105 and have little reason to upgrade further (sample chainset price* £120)
The main advantage of the Ultegra range is that it has a small weight gain over the 105 range, and a possible slight performance benefit (sample chainset price* £135)
King of the Shimano range, and too expensive for many of us, you will need a top of the range bike to make the most of the Dura-Ace components. There are smal benefits in most areas - including weight and gear changing smoothness (sample chainset price* £230)
Note also the new Dura-Ace Di2 equipment with cables concealed in the bike frame and, most important, a small battery driven motor to assist gear changes. This ensures that the chain is always perfectly placed. Expensive but well established among enthusiasts and professional riders.
Within the Shimano range, the components can be mixed and matched to some extent - so a bike can have parts from more than one of their ranges. Likewise, it is possible to upgrade part of the system without needing to change all of it - so in theory you could buy a bike with the cheapest range of parts, then slowly upgrade until you had all top of the range equipment.
In reality you are unlikely to do this because a low-cost bike with top of the range components would be less ideal than a mid-range bike with mid-range components.
* sample chainset price, given as an indication of the range - the price given is for the cranks and front rings on a triple ring