Today most sound systems use transistor amplifiers for economic reasons, but Valve amplifiers remain popular for guitar amplification and for "high end" hi-fi systems.
Valve amplifiers are widely, but not always correctly, associated with the valve sound. In fact this sound has more to do with the circuit topology and circuit design of the amplifier, than to the use of valves rather than transistors as the active gain devices.
Fundamentally, valve amplifiers are amplifier built using valves as the main active components. However, today, valve amplifiers (at least for Hifi applications) are mostly the preserve of extremists and consequently the subject is hotly debated among its enthusiasts, sometimes on the level of religious schisms.
Several subsets of enthusiasts consider that "pure" valve amplifiers shall not use anything except valves as active devices. Others in contrast will use valves for the audio circuit, but will accept the use of silicon gain devices in the power supply or as constant current sources etc.
Other schisms concern the use of triodes vs. tetrodes and pentodes, and the use of directly heated valves vs. indirectly heated valves.
What most valve enthusiasts will agree on is that valves "sound better" than transistors, although there is less consensous about the explaination for this. Some of the explainations offered (for example the use of little or no negative feedback can also be applied to (unusual) circuits built using transistors or MOSFETS, and such circuits in many respects offer many of the characteristics claimed as "valve" advantages, albeitbeing atypical applications for these devices and often introducing other serious problems.
Many of the explainations relate to the circuit topologies pioneered using valves, and traditionally associated with them ever since, regardless of if they are built using valves today, notably the single ended directly heated triode amplifier circuit, which operates in Class A and often has no negative feedback, and this topology is a classic source of the valve sound.
Early amplifiers were by definition valve amplifiers since the transistor did not become common in consumer amplifiers until the late 1960's. The very earliest ampifiers usually had single ended topologies with the most basic type of valve, known as a triode. An audio amplifier using this topology will always be in class A. Class A single ended triode ampliers (Known as "SET"s) have a characteristic distortion spectrum, a simple and monotonically decaying series of harmonics, dominated by modest levels of second harmonic distortion. Second harmonic distortion is musically equivalent to adding the same tone one octave higher, to form a chord. in this case the added tone is at a lower level (typically 5% or less at full power in a no feedback amplifier) but the effect is to "fatten" the sound.
Another aspect of the "valve sound" is that early valve amplifiers often had only limited bandwidth, in part due to passive component technology available at the time, notably capacitors. It must be remembered that he source material available at this time (usually vinyl records) lacked the definition of todays CD's. There was no electronic music with extreme bass, and high frequencies mostly contained noise and distortion, so it was advantagous to not emphasise these.
Transistor amplifiers are almost always Class AB push pull, and this topology tends to cancel even order harmic distortion products. The resulting distortion is therefore dominated by odd order harmonics, which to human sensibilities sound "harsh" etc. Transistor amplifiers made during the 1980's typically also had extremely high gain, but poor open loop linearity, and relied on large amounts of negative feedback (NFB). Some consider that NFB does not sound "natural" or "musical", due to errors in the way it reacts to transients. These errors also reveal themselves in very complex distortion spectra, that humans find "discordant".
In contrast audio valves typically have only modest gain, and are extremely linear. This makes it possible to design very simple valve circuits that rely on this inherent open loop linearity and have little, or indeed no, NFB, and thus have very simple distortion spectra.
The subject is further complicated by the way triodes and MOSFETS have certain similarities in their transfer characteristics, whereas later forms of the valve, the tetrode] and pentode have quite different characteristics, in some ways reminicient of the transistor.
All these factors of contribute to the "valve sound" for HiFi applications. Valve amplifiers however are also widely used for electric guitars, in this case for quite different reasons : the way they distort when overdriven is quite different, again leading to a very distinctive sound.
In more modern times, transistor amplifier have become dominant, mainly because they are cheaper to produce, and operate on lower voltages (including batteries, eg applications such as car radio's.) However valve amplifiers have retained a loyal following among hifi fanatics, especially for SET's, in Japan, and in recent years there has also been a resurgence in the west.
Using modern passive components, and modern digital sources and wide band speakers, it is possible to build valve amplifiers that have the characteristic wide bandwidth and "fast" sound of modern transistor amplifiers, including using push pull circuits, class AB and feedback. And some enthusiasts have built amplifiers using transistors and MOSFETS that operate in class A, including single ended, and these often have the "valve sound".