Compression is a very complex field that you can get in to either a little or a lot or anywhere in between. There are companies whose only job is to do high quality compression for major releases. Part of the trick is that there are different compression algorithms, different storage formats, different codecs and each codec has its own set of settings.
Starting at the most broad, we have different algorithms. A compression algorithm is a particular approach to trying to reduce the amount of information that needs to be stored to reproduce a signal. Examples of compression algorithms are things like MPEG, MPEG-2/H.262, MPEG-4, MPEG-4 AVC/H.264. It defines how to simplify the information and what kinds of patterns are being looked for in the signal.
Then, similarly broad, we have storage formats. These are either physical formats such as Bluray disk or DV tape or file formats such as WMV, MOV, M4V and simply store information that is either raw or more likely compressed with a compression algorithm via a codec.
A codec is a particular implementation of a compression algorithm. It is an encoder/decoder. With standards based codecs, it is not necessary for the same exact codec to be used for decoding as is used for encoding, though proprietary codecs do exist that require the same codec to be installed for playback to function. Part of the complexity of compression is that each codec makes it's own implementation of even standard algorithms and has it's own settings available to adjust how encoding is done.
Often, the better the codec, the more control it gives over how compression is performed. Since settings differ from one codec to another, the impact of settings can similarly differ. There isn't really any substitute for experimentation when learning how a codec behaves, but some key terminology stays consistent between different codecs and even different algorithms.
The most obvious, significant and consistent one is probably VBR (1 and 2 pass) and CBR. Almost any motion based compression has these options. CBR is Constant Bit Rate. It means that a certain amount of data should be used per second every second regardless of the amount of activity or difficulty of compressing. In motion based video, it leads to really high quality when things are relatively still, but quality suffers during times of high motion. It also gives very easy to estimate file sizes.
VBR, on the other hand, stands for Variable Bit Rate and as the name implies, the bit rate can change based on the difficulty of compressing the current scene. Typically a target bit rate and a maximum bitrate are give. The target is what the codec will try to average out to and the maximum bitrate is the level it isn't allowed to exceed at any time. 1 pass VBR will attempt to do so with no prior knowledge of the video being encoded. 2 pass VBR on the other hand makes a pass through the video to determine the characteristics of the video so that it can properly spread the data rates around to achieve the best possible quality with the allotted bandwidth. The downside is that it takes roughly twice as long to encode.
Another relatively consistent concept is keyframes. Many compression work by only actually storing full images occasionally. The rest of the frames are approximations based on the previous full image. Keyframes can determine how often those full frames are used. More advanced codecs will even allow for the actual sequence of frame types being used to be controlled. (For example, with MPEG-2, I, B and P frames are the 3 main types of frames.)
When using non-standard formats, resolution is also an option for compression. It can greatly reduce the amount of information that needs to be stored, but also greatly reduces the quality it is designed to be played back at. When trying to figure out the best compression for the web, it's important to consider both the bitrate and the resolution. Trying to store too many pixels with not enough bitrate will result in compression related artifacts, but storing too few pixels will result in a universal loss of resolution (but less artifacts).
While each codec can have guidelines for approximating bitrates needed for various resolutions, there unfortunately isn't much substitute for experimentation. A video of watching paint dry is going to take substantially less bitrate than Uncle Billy Bob's shaky footage of a high speed soccer match. In general, for motion based compression, the amount of motion in the scenes determines how well it compresses and you can develop a feel for it with time working with a particular codec.
As with your other question, if there are any more specific topics you are stuck on, feel free to ask here or to check a particular format on Wikipedia or similar for much more in-depth information than can easily fit in a Q/A format.