Have a look at the excellent mushroom anatomy diagram to the right. This is a meta-mushroom, containing distinguishing features from many different "classic" mushrooms. These features are what you, the forager, will be distinguishing in your process of making a mushroom identification.
Were I to list all the possible differences between all the thousands of documented mushroom species, this website would be encyclopedic, you would be bored and glassy eyed, and I would have permanent writers cramp. I actually have a highly detailed handout that lists many unique qualities for each major physical mushroom feature. The handout lists 207 distinguishing features and is poster-sized. Even the most dedicated mushroom hunter probably considers the idea of memorizing that level of detail an arduous and uninteresting ordeal. The important part of mushroom ID is being familiar with the major physical features; knowing what your guides are asking you to look for, and how to distinguish them on the specimens you discover.
In addition to physical appearance, there are many other mushroom features that are important to scrutinize in the identification process. Speaking from experience, some fungal groups can only be identified to the genus level if you stop at physical appearance. Often, you will need to discern characteristic odors and even tastes. For some genera, it will be important to examine cap firmness; how the stipe fractures when bent; whether the cap separates cleanly from the stem. You may need to observe color changes after cutting or bruising the cap or stem. For certain groups that are easily confused, it may be necessary to determine the color of their spores. You often will need to observe whether the mushroom was single, clustered, or single but with neighbors in some particular pattern (fairy rings). And, of course, you will need to observe habitat and substrate (what it was growing on).
In the following section, I'll show examples of some of the major features you'll need to be familiar with. In section 2, I'll pull the identification process together, and go through some identification activities.
Please note that these features will not be covered comprehensively. The following is intended to give you a general idea of what you'll be looking for...