Flight under Visual Flight Rules may only be undertaken (in the U.S., at least) if the weather meets certain minima. The conditions that must pertain for VFR to be possible in the United States are laid out in Title 14 CFR § 91.155. As of November 2012, those minima are as laid out in the following diagram:

                         Class A - IFR ONLY - NO VFR PERMITTED
18,000 ft MSL ----------------------------------------------------------------- 18,000 MSL
                  Class E/G - 5 SM (statute mile) visibility, "111" clearance
10,000 ft MSL ----------------------------------------------------------------- 10,000 MSL
                             |                                 |   1 SM vis
                             |                                 |   "152" cloud
                  3 SM vis   |           3 SM vis              |   clearance
 1,200 ft AGL     clear of   |         "152" cloud             |--------------- 1,200 AGL
                  clouds     |          clearance              |   1 SM vis
                             |                                 |   clear of
                             |                                 |   clouds
Surface       ----------------------------------------------------------------- Surface                  
                  Class B        Class C/D/E & Class G night     Class G day
     "111" cloud clearance = 1,000 feet below / 1,000 feet above / 1 SM horiz. 
     "152" cloud clearance = 1,000 feet below / 500 feet above   / 2,000 feet horiz.
Although this table is a bit complex (especially if you're trying to remember it!) its structure is logical. Let's take it from the top down.

Class A airspace - at and above 18,000 feet MSL (flight level 180 and up), no VFR flight is permitted. All aircraft must be operating under IFR to ensure air traffic control is effective.

Between 10,000 feet MSL and 17,999 ft MSL, VFR flight is permitted. Over the US, aircraft will be in one of two airspace types at these altitudes - Class E or Class G. In either case, it is expected that faster aircraft will be operating at these altitudes. Therefore, the visibility requirement is greater - 5 statue miles - and the cloud clearance is more conservative. The "111" clearance stipulates that aircraft must remain 1 statute mile clear of clouds horizontally, as well as 1,000 feet clear of them vertically. This gives more space for aircraft to see each other, as they're likely moving at high speed.

Below 10,000 MSL things get more complex, because there are more classes of airspace that the aircraft might be in. Class B airspace, which surrounds the busiest airports, has what at first blush seems a paradoxically lower visibility requirement - 3 statute miles but aircraft must only 'remain clear' of clouds, without any minimum separation distance. This can be explained easily - in order to operate in Class B airspace, which is the most restrictive of VFR airspaces, all aircraft must receive specific clearance from the controlling ATC facility and remain in communication with them at all times while operating within the Class B. Thus, the assumption is that ATC will be ensuring traffic separation, and aircraft may need to be routed near clouds (by ATC) to conserve space within the Class B circle.

Class C, D, E and Class G (during night time) all share clearance requirements. The most common, "152" clearance - 1,000 feet below, 500 feet above, and 2,000 feet horizontal cloud clearance. The visibility requirement is the same - 3 statute miles. I should point out that 'visibility' does not necessarily mean 'no precipitation'. For example, if a rain cloud is 1,500 feet above you and raining, but you can see for 3 miles through the rain, then you are still in VFR conditions.

Class G daytime (the least restrictive controlled airspace type, and nearly all of the CONUS is Class G or higher) has less restrictive rules, and they differ depending on how far above the ground you are. This permits the use of airports in Class G airspace even when their local conditions are slightly worse than general VFR, to permit departures, arrivals and traffic pattern practice in less-than-ideal conditions. Since traffic in the traffic pattern should be separated by no less than half a mile to a mile (a mile is better) then so long as you have 1 mile visibility, you should be able to take off and land legally. Note that the difference in the minima for Class G occurs at 1,200 feet above ground level (AGL), not MSL.

(IN 5 23/30)