There are many ways to unblock drains.
General recommendation: act before the blockage is total, as a total block makes getting unblocking chemicals to the blockage difficult to impractical.
The most common drain blocking materials are: fat, hair and matted fibres which can be cleared by caustic soda, rodding, or physical removal with hand or sticks.
I can strongly recommend caustic soda (Sodium Hydroxide) as fastest and cheap unblocking solution.
Caustic eats fats, which are the most common blocker of kitchen sinks, but I found it highly effective against all other major blocking materials like hair and matted fibres etc. Hence, it is very useful when it comes to unblocking bathroom sink and bath tube drain pipes.
Boiling caustic is much faster than cold caustic. I never use direction for use which is given on package of caustic soda - it says to dissolve 500g of caustic in 5 litres of cold water, mix it, pour the solution down drain, leave for 30 minutes then flush with water (preferably hot).
Instead, I would carefully open the tub, release volume equivalent to about two table spoons of caustic soda pearls straight into the drain and gradually pour entire boiling kettle of water in to the drain hole. Please be aware that it is important to pour no more than about quoter of a cattle at time and move out of reach of possible splashes, as the reaction of caustic and hot water is quite strong and if you put too much water at one time there could be effect of a small volcano. By the time of pouring the remaining water from the kettle the drain should be unblocked, just finish off with flushing hot water (e.g. form the tap) for about 0.5-1 minute to finish the procedure. This method is not best suitable for blocked toilets.
The whole process of unblocking a drain with caustic soda should not take longer than 3-4 minutes.
In UK you can buy caustic soda in many local hardware/convenience stores or chain DIY supermarkets like HOMBASE, B&Q, Wilkinson.
Caustic is a high risk chemical. It burns skin and can cause serious eye damage.
If you get any at all in your eye, it is important to seek treatment urgently. Delay in treatment (it doesn't hurt) generally results in serious irreversible damage.
Caustic soda is a strong alkali.
Never mix a strong alkali with an acid.
Eye protection recommended.
More information about ways to unblock drains: http://wiki.diyfaq.org.uk/index.php?title=Unblock_a_Drain