What geographical area does a METAR cover?
Geographically METARS are for a single specific location (the station identified in the report).

An automated METAR with no human supervision covers the conditions within approximately 6 inches of the weather station reporting it (for example, a badly positioned AWOS/ASOS station may be shielded from wind in some directions which affects what it reports).

A manual (or manually augmented) METAR covers the conditions the weather observer can physically see (clouds, precipitation, visibility, etc.) in addition to the ones they can measure (temperature, dew point, barometric pressure), which are valid for the point of measurement. 



Practically speaking if the weather station is positioned well a METAR (like a TAF) is probably fairly representative of conditions within 5 miles, assuming there are no geographic features such as mountains which may mess with the weather. The data is still still "useful" within about 20 miles (e.g. KMTP uses the weather reported by KHTO, about 17 miles away), though you should obviously cross-check it - in that example the winds at Montauk (right on the water) can be noticeably different from the winds at East Hampton (further inland).

Some information is broadly useful over a wider area. FAR 91.121 requires you to set your altimeter to the value reported by a station "within 100 nautical miles" of your position when you're operating below 18,000 feet because things like barometric pressure tend to affect relatively large areas.