The way a mushroom's gills are attached (or not attached) to the stalk is very important in identifying the mushroom. The numbered diagram gives the most standard types of gill attachment.
Freegills are those which do not reach the stalk. This is the only kind of gill attachment which is not really an attachment.
This is shown as #2 in the numbered diagram.
If the whole gill extends down the stem, it is called decurrent. The edge of a decurrentgill is usually not horizontal, but approaches the stem at a slant, so the gill constantly gets wider as it approaches the stem.
This type of gill attachment is #3 in the numbered diagram.
Subdecurrentgills run parallel to the cap for most of their length, then the edge suddenly bends down the stem just before it gets there.
This attachment is #4 in the diagram. It is the last type of attachment on which there is general agreement. The rest of them are the way I use the terms in this website, and are quite reasonable, but it will be easy for you to find people who disagree with me. For a discussion of why this is so, see the entry for natural language.
An emarginategill has roughly the same height for most of its length, then suddenly becomes much shallower just before reaching the stem.
This attachment is #6 in the diagram.
Sinuategills are just like emarginategills, except they curve back down the stem for a little bit just before attaching.
This attachment is #7 in the diagram.
Sinuate is also sometimes used to describe a long edge, like the edge of a gill or the cap, which is smoothly wavy, like a series of swells in the ocean. The margin of the cap of this Phellodon tomentosus is sinuate.
Drawing #8 in the numbered diagram illustrates an attachment that is fairly common, seems to me to be quite distinct from any of the others, and has no specific name.
It seems to me that some people call this adnexed, and others call it emarginate.
Although gill attachments are very important in identifying mushrooms, one should not think that they are a hard-and-fast characteristic. A species may feature several different types of gill attachments, especially the three narrowly attached types ( adnexed, emarginate, and sinuate), even on the same fruiting body. The term notched is sometimes used to mean either emarginate or sinuate, as in the cross-sectioned fruiting body shown in this picture.
One should also be careful to take the total shape of the fruiting body into account. From the side, the gills of this mushroom look very decurrent, but when you look at it in cross-section, you see that the gills only slant towards the stem because the cap slants so much. So the gill attachment here would just be adnexed or subdecurrent. I'm sure that someone somewhere has listed it as decurrent, though!
Another phenomenon to take into account is that as the mushroom dries out and the cap lifts, gills that were attached to the stem sometimes secede: they pull away from it. The gills on the upper right of this picture are subdecurrent, as they ought to be. The gills on the lower right snapped away from the stalk, with an audible pop, as I was turning the mushroom over to take this picture.
Likewise, the gills on this mushroom used to be adnate, but they have pulled away from the stalk as it aged. This is one reason to gather several specimens when you make a collection of a mushroom: if your old mushrooms have " free" gills but your young ones do not, you will know that the old ones are really seceding.