It's like filling up a bucket with water. If you don't turn the water on very high and it's just trickling (f22) then you have to leave it on longer (slower shutter speed) to fill up your bucket (your correct exposure). If you crank up the water (large aperture like f2.8) then you don't leave the water on as long (fast shutter speed) to fill up your bucket (correct exposure).

Now, notice that each shutter speed is half (or twice depending on which direction you're going) of the next shutter speed. This means that 1/30 will let in in twice as much light as 1/60. Now, your f-stops work the same way. Each stop is half (or twice depending on which way you're going) of the next one. So f4 lets in twice as much light as f5.6. So if you take a reading of 1/60 @ f4 and want to use a smaller aperture for more depth of field you can go to f5.6. But now that you've closed the aperture down one stop you have eliminated half the light. So to fill up the bucket (correct exposure) you have to let it shine on the film for twice as long. So you have to use a shutter speed of 1/30. So your new exposure is 1/30 @ f5.6.

Many films have a scale of aperature and shutter speeds. Most films have the sunny 16 rule and settings shown on the inside of the box.

(1) "Reciprocity" is the ability to trade aperture f-stops for shutter speed stops and have the same exposure on film. "Reciprocity Failure" is the breakdown of this relationship. George is quite correct that it's not an issue with long shutter speeds until you get to about one second exposures and longer. However, the specific shutter speed at which reciprocity failure occurs, the effect it has, and how to compensate for it is different for each specific film. If you're contemplating long exposures, read the film data sheet. It's not that difficult with B/W films. The compensation required simply makes the exposure time a little longer. It can be quite difficult with color films as failure kicks in at different exposure times for each emulsion layer resulting in color shift. In addition to adjusting exposure time, there are often color correction filter requirements too. In addition, reciprocity failure also occurs with very fast shutter speeds. This isn't as much to worry about now as it was about 30-40 years ago. All the data sheets I've read for current films (and it's quite a few) show 1/10,000th second at the short end. This is much higher than it was for some films when thyristor controlled auto-flash units were introduced in the 1960's. Look at the spec sheet(s) for your flash unit(s). There should be a minimum and maximum flash duration listed (duration is how the flash adjusts how much light it puts out when under "auto" control; not brightness of the tube). It can also be a problem if you're doing special high-speed photography with a rapid firing strobes.

(2) The lens aperture f-number system provides a common system applicable to all focal lengths. The f-number is the focal length of the lens divided by the effective aperture diameter. If you have a 50mm lens set to f/4, the effective aperture diameter inside the lens is 12.5mm. The amount of light allowed through a lens is directly proportional to the area of the aperture opening. This means it's directly proportional to the square root of its diameter. Thus, the f-numbers, which are based on aperture diameter, change with a factor of the square root of two (~1.414). The common f-numbers are rounded-off approximations. That's why changing from f/8 to f/5.6 lets in twice as much light (divide 8 by 1.414). Changing from f/8 to f/4 lets in four times as much light; twice the diameter is four times the area.