Dōzuki are single-sided saws with a stiffening rib. These may be fitted with rip teeth, or cross-cutting teeth, and are, because of the rib, not intended for deep cuts. The rib improves blade support, so these saws are a better choice for shallower precision cuts. Typically these are sized between 150mm and 300mm. A bigger saw size generally means bigger teeth and reduced TPI count, however custom saws can be ordered which are large in size with smaller teeth and high TPI as well. If you want to cut carcase dovetails, then a dōzuki rip saw is what you want, though what have been often sold here for that purpose are dōzuki crosscut saws.
Ryōba, a word which literally means "two-leaf", are double-sided saws. Ryōba are intended to be more multi-purpose in general, as one side has rip teeth, and one side had cross-cut teeth. If you were only going to have one saw, then this is likely the logical choice, and your only decision, besides price point, is size. They are sized from about 150mm to 330 mm generally. For timber work, you would be looking at a 270~330mm saw, for furniture making you might want something in the 210~240 zone, and for instrument making, say, 150~180 might be more ideal. Depends upon the size of the sticks you work with. The most 'all around' size is probably a 240mm.
Single-sided saws without the stiffening rib are termed kataba and essentially they are single-sided versions of ryōba. These are available in the same range of sizes as for ryōba, and can be had with either rip teeth, or cross-cut teeth, or with teeth shaped specifically for angled cuts. One advantage to kataba is that with really deep cuts where the saw is entirely buried in the wood, there is no rubbing and tooth wear, and potential marking of the cut face, going on with the un-used side of the blade. Also, with some types of kataba, the total height of the blade is a bit less than it might be with a ryōba version of the same tool. If your work was of a specialized nature where a certain size of saw was almost always going to be used only for ripping, or only for cross-cutting, then a kataba with the required tooth pattern makes sense for that application.
There are additionally saws for cutting into narrow spaces, saws for detail work in close confines, saws for flush-trimming of pegs, etc., and so forth. With a given type of saw, say a 270mm ryōba, one can obtain models with a thinner saw plate or a thicker saw plate, minimal tooth set or maximal tooth set. In general, thinner blades and minimal tooth set are better choices for experienced users, while thicker saw plates and greater set will suit a beginner better. A lot of these options won't be apparent to consumers of Japanese saws outside of Japan since only a minority of products available over there are imported/marketed. Custom made saws are obtainable from Japanese dealers if you have something particular in mind that can't be found on the commercial market, though there will be a wait and the cost is likely to start around $300. For casual users, and even no shortage of pros, the impulse-hardened teeth seen with mass-produced saws work quite well for most users.
