Most straight razors are hollowground. What this means is that the sides of the razor are hollowed out by grinding on a wheel. You can press the blade down on a flat surface and the spine and edge make contact, but the hollowgrind part in between does not. This hollow is very pronounced, and obvious to feel and to the eye. The hollowground razor was a great technological leap forward in the history and production of the straight razor. Hollowgrinding makes a razor much easier to hone.
The wedge razor is the older style of blade. As the name suggests, the sides are flat and the blade has a straight wedge taper from spine to edge. These razors are a bit more difficult to hone properly and I suggest that beginners avoid them until they have more experience. Myself, I do not like them but they have plenty of fans.
A hollowground razor can have a rather deep and high hollow, or a more subtle hollow. We often refer to varying degrees of hollow as quarter hollow, half, three quarter, full, or extra full hollowground. What they all have in common is that the spine rests on the hone when honing, and serves as the bevel guide to keep the bevel angle very consistent through the honing process, resulting in a very flat and precise bevel.
Wedges and near wedges do not enjoy that advantage. It is true that the initial grinding involves the entire face of the razor, and on rare occasions one might basically regrind the wedge razor by honing flat on its face, but normal honing calls for lifting the spine slightly. This ensures that the edge gets good contact with the hone, and the contact area is fairly small. Also wedge blades typically have a very acute natural bevel angle. In other words, the spine thickness is smaller than a hollowground razor of the same blade width. This was once done freehand, often with a fingernail used as a honing angle guide. Now, we almost universally use electrical tape on the spine to artificially and temporarily thicken the spine to raise the back of the razor to the desired honing angle. This sets the working bevel, or primary bevel. Initially this bevel surface is very narrow but over the years as the razor is used and honed, the edge is ground back into thicker steel and so the bevel surface gets wider. The wider bevel surface is slower and more difficult to hone properly, and so a micro-bevel is sometimes applied, by adding another layer of tape or two for only the last few finishing laps. This microbevel of course has a very narrow surface area, and hones very easily and quickly. The tape doesn't get a chance to wear or compress, so this bevel remains very flat and precise, and renders a sharp edge quickly and easily.
Refreshing this compound bevel edge is done with the extra layer or layers of tape, and ideally only requires a few laps on the finisher. It is important to keep wear down and preserve the edge geometry as long as possible. Eventually the microbevel expands to the full width of the primary bevel. When this happens, the primary bevel ideally is reset, and a new microbevel is formed at the finishing stage of honing.
If all of that sounds very complicated and involved, well, it is, compared to hollowground honing. But done properly it does usually result in a very good edge that is capable of shaving very effectively. Some fans of this type razor appreciate the extra heft of the blade, or the rigidity compared to a hollowground razor. Some are drawn to it by tradition. Wedge razors are yesterday's news and have been succeeded by the ubiquitous hollowground razor, but they are still around, and still being made, and still have a market. And so the newbie to honing and shaving with straight razors will likely encounter one, sooner or later.
I am not one of those who likes wedges, as I said before, and I consequently have little expertise in their honing, so I will not write a tutorial on them, preferring to let those with a greater interest in them do that. I will say that a newbie should initially avoid them, and maybe try one later, after gaining some experience. You might like them, you might not. Just be aware of the difference.
Occasionally you might encounter a hollowground razor that has been honed so much over its lifetime that it begins to assume the form of a wedge. Such a razor should be honed in similar manner to a wedge. If the spine thickness is such that the bevel angle is too obtuse for this, then hone it flat with more pressure toward the spine than the edge, to reduce the natural bevel angle, then hone as a wedge. Or toss it. Or give it to someone who appreciates razors in this state. If the bevel surface at the edge is too wide, it is not going to be easy to hone without tape and a compound bevel.