Almost all Tube amps have at least two or more transformers. The Power and Output Transformer are the primary two used in tube amplifiers. So how much do these components contribute to the sound of an amp? The Power Transformer does exactly as its name implies, it provides the power needed to run the amp. For the most part, a properly chosen power transformer has little or no effect on the tone of the amp. "Note that I said properly chosen transformer". A poorly or underrated power transformer can have an effect on the sound and since it is one of the most expensive part in the amp, many manufactures skimp on this transformer and therefore changing it out can make a big difference. A rule of thumb is that the power transformer should have a VA rating of at least twice the output power of the amp. This insures that the amp will not be starved when pushed hard and is will stay tight all the way to 10. Now some players actually prefer some sag when the amp is pushed hard. I'm sure all of you have heard that many tube amps only sound good when pushed hard and the main reason for this is this sag. When the transformer can't provide the necessary power to keep the power capacitors charged, the voltage drops in the power supply and caused the output tubes to start clipping therefore changing the sound of the amp. Of course that there are many other variables that can contribute to the tone difference at higher volumes but power supply sag is the biggest contributer of this effect. For hard hitting metal and hard rock, this sag can sound farty and thin especially for lead playing. A higher power rated transformer helps the power supply capacitors stay charges and able to produce a more linier sound all the way to 10. So if your amp tends to fall flat on its face when turned all the way up. A Power Transformer replacement may be in order. If the amp sounds pretty much the same all the way to 10 then its a good bet that the power transformer is properly sized and there is no need for replacement.
Unlike the power transformer, the Output transformer can play a huge roll in how the amp feels and sounds. There are dictionary sized books that cover all the aspects of how output trannies operate so I'm not going to go into major detail about them and just cover the basics. There are mainly two types of output trannies, Push/Pull and Single ended. Pretty much any amp over 25 watts is a Push Pull type since single ended trannies become too large and unpractical above 25 watts. Everything including witchcraft, sorcery and good old luck has gone into designing output trannies but most of the hard work has been focused on improving linearity and efficiency for HiFi amps. Since linearity is not a big concern in guitar amps as most often the case the signal going into the tranny is heavily distorted and clipped. So efficiency and loading options tend to be of most concern. Now I'm sure there are many opinions out there that think that other factors are more important but in my experience and style of guitar playing the prior two factors are more important in shaping to overall sound of the amp. Because there are so many variables that effect the sound of an output tranny, they are more commonly swapped out for different units than power transformers. If you amp lacks low or high end fidelity, loading options or just have a bunch of money to spend then changing out the output transformer might be in order.