A METAR is a french acronymn for "Aviation routine weather report" and can be thought of as a report of what is happening now at the airport. The reports are usually taken every hour unless changing conditions dictate a special report. Metars are broadcast on the ATIS along with other pertinent airfield info such as runways and approaches in use.
A TAF is the french acronymn for "Aerodrome forecast" and is a forecast created every 6 hours for conditions expected during the following 24 hours at the airport.
Most pilots use both reports together to get a completete picture.
Obviously the Metar is most relavent for the departure airport as it tells you what is happening now, and its information is used in take off performance calculations. The departure TAF is also going to be refered to in order to be aware of any approaching conditions that may have an impact.
For the destination, the TAF is more relavent and is the deciding fractor in wether or not an alternate is required and if so, the alternate's TAF (along with that airports' facilities) will determine if it is a suitable alternate. Metars also play a factor here; by looking at a series of them and comparing them with the TAF one can see if the TAF is "telling the truth". Making this comparison enables us to see if the forecast weather is changing faster, slower or, in some cases is completely different from the forecast.
Timing is everything, it makes the most sense to check the Metars just after the hour, in order to get the most up to date as they are usually published shortly before or on the hour. TAFs are published at 0000z, 0600z, 1200z and 1800z.
Once airborne, a check of the destination weather is definitely in order during longer flights, or any time the TAF indicated rapidly changing conditions were possible.
This is just my personal technique but I think most of us use something like this along with checking the wealth of additional information and charts that are available.