"Conifer" is an arboricultural term meaning, literally, a cone-bearer (such English words as "refer" and "aquifer" also use the FER Latin root). Trees that fall into this category reproduce by forming a cone rather than a flower as a container for their seeds. It is this fact regarding reproduction that points us to the difference between evergreens and conifers (see below).
Most conifers are evergreens, but not all of them are.
Examples of conifers include:
Again, don't confuse "conifer" with "evergreen." While there is overlap between the two words, they do not mean the same thing. As you can see from the above, the former pertains to means of reproduction (the cone); the latter, by contrast, refers to a tree's leaves (or "needles").
Perhaps the best-known example of the fact that not all conifers are evergreens is the larch or "tamarack" (Larix laricina. In summer it looks like it would be one of the evergreens, because it bears needles. But this conifer is a deciduous tree.
Likewise, not all evergreens are conifers. Holly shrubs are evergreen, but they are not conifers because they reproduce via berries (technically, "drupes"), not cones.
Beginners often become confused by these terms and end up asking the wrong questions. For example, you'll sometimes hear someone asking if such and such a tree or shrub is "deciduous or a conifer?" as if it had to be one or the other. But that's sort of like asking if Mount Rushmore is "in Europe or in Asia?" It's the wrong dichotomy to set up. From the foregoing, you will see that the difference the person truly has in mind is between "deciduous" and "evergreen."