I'm writing this for a few friends, it's meant to be an entry level primer on tube amp topologies. I'd like to keep things very general, so that a newbie to the hobby can gain some vocabulary for the various tube types and output topologies. There are exceptions to probably everything I will say, but I wanted to share what I could in as general language as possible. There are other threads for hard-core techies, this one is for the novice.
Triodes have one and only one "control grid". Triodes can be used in Single Ended Triode (SET) circuits, and in push-pull circuits. If a true triode is used in any circuit as an output tube, it can not utilize "pentode" or "ultra-linear" mode.
Pentode tubes have three control grids, and the addition of the extra grids allows the pentode to be run in various different output modes: triode, pentode, and ultra-linear. Triode mode mimics a true triode tube, is often of a lower power than the others, but is popular because of it's sonics. Pentode mode usually uses a tube to more of it's maximum output potential, and often gets the highest power ratings out of a tube. It's generally fallen out of favor, but making a comeback in certain DIY circles. Ultra-linear mode is sort of a compromise between the two, with power ratings between triode and pentode. Ultra-linear mode is only possible if the output transformer has a special tap that supports this use.
There are three basic output topologies for tube amps: Single Ended (SE), Push-Pull (PP), and "Output Transformer-Less" (OTL). OTL amps do not use an output transformer, are fairly exotic due to the number and size of tubes requires, and other factors. They are worthy of mentioning, but beyond the scope of a document aimed at a novice. As both SE and PP amps require an output transformer (OPT), the characteristics of that transformer will dictate many factors about the amp. Generally speaking, an output transformer will be either a single ended OPT, or a push-pull OPT. Either PP or SE OPTs can have an ultra-liner (UL) tap. If your output has it, you can optionally use it. One could argue that the output transformer is the most critical part in a tube amplifier.
A Single Ended amp uses a single output tube. If the tube is a native triode (300B, 2A3, etc.) this is a SET amp. You can run a pentode in Single Ended Pentode (SEP) mode as well, but it's not a SET amp. Using a pentode, you can have a Single Ended Pentode (SEP), and if your transformer supports it a Single Ended Ultra-Linear (SEUL) mode as well. It is possible to switch between these modes.
Push Pull uses a pair of tubes to power the speaker. Triode, Pentode, and UL modes are all available here, too, depending on the choice of tube and transformer.
Parallelism is also possible, usually for more power. A Parallel Single Ended (PSE) amp will use more than one tube run in single ended mode. A Parallel Push Pull (PPP) amp will use two or more pairs of outputs. While it's usually easy to spot a Parallel Push Pull amp because of the number of output tubes (4, 6, 8, etc.), the most common Parallel Single Ended amps can resemble a Push-pull amp, as each have 2 output tubes. Generally speaking, parallel amps are not as common as non-parallel. Vintage parallel push pull amps were often PA type amps. Parallel Single Ended amps seem to be more common in modern times, as people want more power while trying to retain some single ended sonic characteristics.
Single ended amps are generally prised for their sonics, but put out less power than push-pull amps. They are generally simpler and have less parts than a push-pull amp. They tend to be fussier about hum and noise, so particular attention must be paid to the quality of the power supply, amplifier layout, careful heater wiring, and other factors that induce noise. In some ways, parts selection becomes more critical because there are fewer of them to begin with. This can work to a DIYers economic advantage too, as sometimes it's possible to use a budget for fewer but higher quality parts. While circuit choice is still very important, due to the simplicity of the design, most SE circuits resemble each other to a great degree.
Push-pull amps generally have more power than SE amps for any given tube. Generally speaking, they tend to be more complex then SE amps, with a higher parts count and more tubes. They have some noise-rejection qualities inherent in their topology, and sometimes this can present less of a problem to get noise free. Choice of circuit can be of particular importance in a push-pull amp, some can really sound better than others. There is also a greater number of circuits to choose from, as different designs handle the more complicated push-pull circuitry in different ways. Parts selection is still important, but perhaps circuit choice is even more critical in a push pull amp.