Electric guitar strings are typically constructed by tightly wrapping one wire (the "wrap wire") around another (the "string core").
Small gauge strings, like the two or three highest-pitched strings on your guitar (G, B and high E), omit the wrap wire, leaving a single, plain wire.
(This post is one of a series discussing the technical aspects of guitar strings.)
Historically the wire or string at the center of a guitar string was round (in cross-section), as you might expect. These are known as round-wound strings.
Since the 1970s most string manufacturers have switched to a different design, one that uses a core wire in the shape of a hexagon (again, in cross-section). These are known as hex-core strings.
The core shape does change the acoustics and the feel of the string, but the difference is pretty slight. Most guitarists would not be able to tell the two apart in a blind listening test.
The real difference is that hex-core strings are easier to manufacture. The hex design leaves a small gap between the core and the wrap-wire - the wrap-wire is hitting the sharp corners of the hexagon, not the flat sides. This helps to keep the wrap-wire from slipping as it is being wound around the core.
That said there are some differences between the two, even after the manufacturing process.
Round-core strings may be higher quality in general. Since round-core strings are harder to make they are often made by hand. If you are going to invest in a craftsman to hand-make each string, you're not likely to skimp on the materials or quality control. That's not to say that hex-core strings are of poor quality, but there probably is some difference between handcrafted strings and those wrapped by machine in a factory.
Over time the gaps between the core and wrap in a hex-core string can fill with dirt and grime that is resistant to cleaning. This makes hex-core strings start to sound "dead" before their round counterparts. Coated strings were introduced in part to combat this problem.
On the other hand, the hexagonal core makes the string stronger and stiffer. This makes them less likely to break and better able to stay in tune.
Conversely, round-core strings are little bit more flexible, giving them a warmer sound with more sustain. This is probably the primary reason for the difference in the way the strings sound.
In general, hex-core strings are thought to have a brighter sound with a strong attack and less sustain. Round-core strings are thought to have more sustain, and a warmer, muddier tone that many consider a "vintage" sound.
That said, it is actually pretty hard to do a real, apples-to-apples comparison of round-core and hex-core strings. The material, wind and method of manufacture are usually different for hex and round core strings. Those factors might have a bigger impact on the overall sound than the shape of the core.
The effect of the shape of the string core may be subtle but the wind of a string has a huge impact on both the way the string sounds and to feel of the string beneath your fingers.
Typically the wrap wire has a round shape (when seen in cross-section). Strings like this are called round-wound and are by far the most common type.
The circular profile of roundwound strings introduces deep grooves between each ply of the wind that you can readily feel with your fingertips. These grooves give roundwound strings a bit of a rough feel that is harder on both your fingers and the frets.
Over time these grooves will fill with dirt, oil and grime from your fingers. That's not a comment on your hygiene, it's unavoidable. A dirty string is less uniform, less flexible and less lively, making the string sound "dead" over time. Cleaning the string may bring it back to life. Many guitarists wipe-down their strings after each play session, which is a pretty good practice with or without roundwound strings. Some guitarists even boil their strings in water but the practice is controversial.
The roundwound profile is more prone to squeaks and fret noise, but proper technique will eliminate most or all of that.
Roundwounds are known for their bright, crisp tone with a fast attack and a slow decay, making them the standard across most modern genres of popular music.
The wrap wires used for flatwound strings are squished into a more rectangular shape, almost like a ribbon wrapped around the core. This shape impacts the sound, feel and even construction of the string.
The flattened shape fills more of the gap between loops of the winding, leaving only small and shallow grooves that are nearly imperceptible to touch. This gives flatwounds a very smooth texture, almost as if the string were made of one continuous ribbon rather than a wrapped wire.
The core string in a flatwound is usually thicker than that of the equivalent roundwound of the same diameter, since the wrap wire is thinner. This makes flatwounds feel stiffer and have a higher tension at the same tuning.
Flatwounds have a distinctive sound, warmer and more muted. Detractors sometimes claim that flatwounds sound like dead rounds. This is true to an extent - like dead rounds, flatwound strings aren't as bright, especially in the treble spectrum - but they shine the middle and low end. The higher tension and relative stiffness reduces harmonic overtones and emphasizes the fundamental note. Flats are not well suited for rapid, finger-tapping guitar solos on the high-numbered frets. A sleek surface and slower attack leads to a smoother transitions between notes, giving flatwounds their characteristic flowing, mellow sound. Flatwound strings are the standard for jazz guitarists, and are popular with bassists as well.
In some ways flatwounds are more "low maintenance". Flatwound strings are easier to keep in tune, last longer, and maintain a more consistent tone throughout their lifespan. Their smooth surface is easier on your fingers and on the frets of the guitar.
Halfrounds are a bit of a compromise between roundwound and flatwound strings, meant to provide the brightness of rounds and the smoothness of flats. They are sometimes called "halfwound" or "groundwound" strings.
The latter name (groundwound) hints at way halfrounds are made: A round wire is wrapped around the string core then ground-down to a semi-flat surface. The part of the winding that makes contact with your finger is smooth like flats but the part that makes contact with the core is curved, like rounds.
Halfrounds more or less achieve their objective, offering a brighter sound than flats and a smoother, less squeaky surface than rounds. But they truly are a compromise - neither as bright as roundwound strings nor as smooth as flatwound strings.
This post is one of a series discussing the technical aspects of guitar strings.