Among many other factors, string material can have a big impact on the tone of your guitar. There are many options out there for electric and acoustic guitars. Today we will try to help give general guidance on what strings may get you closer to the tone you are after. So if each guitar has a range of tones it can offer, selecting the right strings will tap into different areas of that guitar's sonic spectrum. It is also notable that we will be focused on the wound strings as the plain strings are always steel.
Pure Nickel Strings: These strings will have the "warmest" tone and resist corrosion well too. This is a good string to consider if you are a blues player of want to tame the treble of your single coil guitar. They can be a little too "dark" sounding for some players and, in particular, humbucking guitars may lose articulation in some cases.
Nickel Plated Steel Strings: This is the most common electric string out there. The tone is pretty much in the middle as far as treble and bass response go. This is a great place to start and is perfect for the majority of players. Most genres will enjoy the balanced tone produced by these types of strings.
Stainless Steel Strings: Stainless strings are the brightest of these three types and are well suited to guitarist seeking a very articulate and cutting tone with nice even bass response. These are perfect choices for many harder rock and metal players as they will still cut through even in heavily distorted settings. They also last the longest of these three main string types. They are very well suited to many genres but you should really seek them out if you are suffering from an overly "dark" sounding guitar. On a single coil guitar there is a possibility of getting a bit harsh sounding from the added treble response but it depends on the guitar and amp. As a warning though, stainless will wear your frets faster!
There are many other types of materials used by various manufacturers but the majority use variations on the three listed above. There are other factors to consider too (such as the core shape, winding techniques used, and coatings to name a few) but the list above will get you started on the path to the perfect tone.
Bronze Strings: These strings are commonly seen as 80/20 or 92/8 alloys. The tone of these strings is bright and loud. They are a great choice for bluegrass music or any style that requires big, articulate tone. The downside is that they lose their tone pretty quickly from oxidation.
Phosphor Bronze: Probably the most popular and commonly available, these strings have a "warmer" tone with less treble response than bronze but still sound fairly lively. These are also good to tone down an overly bright acoustic. They last longer than regular bronze due to the phosphor in the alloy.
Silk and Steel: These strings have a soft and mellow tone and also have less tension. This can make them perfect for an older parlor-style guitar that is more sensitive to string tension. It also means that they are easier on the fingers than the other types of strings listed here. They can be overly mellow for most people.
Brass: Rarely seen these days, this string is very "bright" and has very little bass response.
Coated: Coated strings last much longer than non-coated strings. They do tend to lose a bit of treble response but not much. These strings are great for a gigging musician as the extended lifespan allows for less string changes. They do cost roughly 2-3 times as much as an uncoated equivalent.
String Gauge on Acoustics: This has a big factor in the tone of an acoustic guitar's tone. The heavier the string, the more volume it will produce. An easy way to estimate is to look at the body size of the guitar. Larger bodied guitars tend to sound better with heavier strings while smaller guitars often will prefer lighter strings. It is also important to note that a heavier player will most likely prefer a heavier string while a lighter player will use a lighter string. This is very general and every player's needs will be slightly different.
Overall, you should experiment to find the perfect set for your guitars. Every guitar is different and every player has individual needs. Good luck and have fun!