In the context of guitar for example:
However when I play them they have a different sound and has a different frequency. What makes the G note a "G"? What makes notes share the same name?
Historically, two pitches whose frequencies are in a 2:1 ratio (or 4:1, or 8:1, etc.) are considered "the same". This is known as octave equivalence, and it based on how well frequencies in that ratio blend together.
In the twentieth century, it was standardized that 440Hz would be the defining pitch around which others would be tuned. 440Hz was assigned to A; thus, 220Hz and 880Hz are also A.
Pitches are often distinguished using "scientific notation", which given middle C the designation C4. The C one octave higher would be C5, and so forth.
See also What is the origin of the notation A4, B3, F5, etc. (i.e. ), which also discusses the origin of A440.
So the G in the question is defined in relation to A-440, as are all of the other notes in the chromatic scale. The exact definition of G (or even A for that matter) depends on the particular tuning system being used.
Also of possible interest: Whereabouts is G?
There are many questions/answers on this site related to defining pitches and understanding various tuning systems. Starting by looking through the list of questions related to this one would be a good starting point.
There are 7 note names designated in music - 8 if one follows the German pattern. A B C D E F G (and H). They're totally man-made, having no real scientific association.
Taking any one of them, it has a pitch which is designated to its name. It was not always standardised - centuries ago, one country's (or even town's) G would maybe be different from another's.
At those times, when musicians, and others, didn't travel far, it wasn't a problem, And in any case, most instruments could be re-tuned to match the pitch of the new venue's other musicians anyway. But standardisation came along, and a lot of countries agreed that A=440Hz was a good compromise, as Aaron states. Still not exactly true, as some orchestras in the world prefer to deviate slightly from that 'industry standard'.
So, given that A=440Hz, all other notes, whether simple letters, or sharps/flats thereof, have their own pitch, therefore frequency, related to that. Any note could have been chosen as the datum point, but it was A.
Your G, then, has its own frequency, and it happens, physically, that any notes with double, quadruple, or even half of that frequency, will sound 'the same', albeit higher (or lower). Consequently, they will also be called G. It's interesting that the same pitch G (say G5), played on different instruments will sort of sound the same, but at the same time, different. That's due to the overtones or harmonics that every instrument produces. Some of those harmonics are the octaves of G5, along with other notes, which gives each instrument its own timbre. But, again, in any G note, there will be other G octaves discernable.
Sound is created by vibrating thing (in physics we’d say oscillators). Such a thing could be a vibrating membrane or a vibrating string. Here we have certain wavelengths that are in some sense compatible with the string or membrane, resulting in standing waves, while other wavelengths are blocked out pretty effectively. If we look at a string you will see that these wavelengths are more or less integer parts of the length of the string, so once the length of the string, half of the length, one third, ...
In terms of frequency this means integer multiples of a base frequency. We then talk about a fundamental pitch and harmonics. In fact these harmonics are not always exact integer multiples of the fundamental (we call this inharmonicity), but usally we get the strong notion that alongside with one frequency such multiples are produced.
This also means that if you take a note and a note with twice that frequency (higher) you will find that the frequency spectrum of the higher note is in fact part of the spectrum of the lower note: If the lower note has frequency f and the higher one 2f then the harmonics of the higher one are k(2f) = (2k)f, so exactly the even harmonics of f. In fact if both notes are produced at the same time we cannot exactly tell if this is one note or two notes (consonance). Thus the brain has learned that these two sounds are very similar, in fact so similar that we kind of recognize it as the same sound, just higher and lower.
This form the basis for a system of naming pitches: We separate the information of the height of the pitch as the octave of the pitch and then give that which remains a name. Not all music uses all possible pitches. Our pitch system was more or less deduced from gregorian chant and medieval music theory (which is kind of based on a wrong concept of ancient greek music theory), which gives rise to seven basic pitch names (A, B, C, D, E, F, G). Gregorian chant had two different types of B, a high B (B durum) and a low B (B mollum), which corresponds to modern B and Bb (in german H and B). The symbols used for this then gave rise to modern accidentals (b from B mollum, # and natural sign from the B durum). These can be applied to any pitch to create alterations of these pitches to fill gaps between them.
Ignoring the big complex topic of tuning theory we thus arrive at our modern pitch system where each pitch is denoted by an octave and one of 12 base pitches (each of which can be spelled enharmonically in infinitely many ways).
We, as human beings, devoted either names or letters to certain frequencies and their related ratios. So, for example, if 20 musicians get together and announce that they will call 440 Hz Sponge Bob, and if most people follow them, then this frequency will become Sponge Bob note. Daniel Levitin briefly expresses this in his book This is Your Brain on Music.
On this page you can look up letter names devoted to frequencies. When you play the same note an octave higher, the frequency doubles (approximately). I said approximately since the instruments are not perfect; you get the half value if you play an octave lower. In the end, the sound preserves the quality and what you hear is a distinguishable G note.
Boethius devoted letters to musical notes in the 6th century; we still use them, thanks to Boethius.
The thing that makes you hear a note are sound waves. The guitar string vibrates the guitar body which vibrates the air molecules all around us. The vibration in the air is in the form of a "traffic jam" of air molecules: Some molecules start moving fast (because they were bumped by the moving guitar string), these bump into molecules moving slow; fast molecules (from the guitar string) are again coming in hot from behind, and so on as long as the string vibrates.
The vibrating air molecules bump into your ear drum, which vibrates some tiny bones in your ear, which excite some brain neurons.
Anyhoo, sound is all about vibrations. The simplest forms of vibrations are periodic. "Periodic" means that the vibration doesn't "change" in some sense, in the way your metronome goes back and forth and back over and over in the same monotonous fashion. When you pluck your guitar string, the vibrations aren't perfectly periodic (the note fades off as you let it ring), but it's pretty close. It turns out that when you pluck a particular guitar string, and it starts moving back and forth and back - then just like your metronome, the rate at which it goes from one side to the other and back of its vibration does not change, even if it gets softer over time. This is kind of magical, and you should take a moment to let that sink in.
This factoid is what makes a particular string on a guitar ring at a certain "note". The "note" is the "period" of the vibration of the string. The time it takes for the string to go from one side to the other. In the case of A4 on a piano, the time it takes the string inside the piano to go from one side to the other and back in its vibration is 1/440th of a second. This is what we mean when we say that "A4 is 440Hz".
The next A note above A4 is A5 at 880Hz. This means it takes 1/880th of a second to go from one side to the other.
Here's the kicker: If A5 takes 1/880th of a second to go from one side to the other, then it takes 1/440th of a second (i.e. twice 1/880th of a second) to go from one side to the other and back and to the other and back. So, if we pluck A4 and A5 at the exact same time, then after 1/440th of a second, the strings are both back at the same position. After another 1/440th of a second, they are back at the same position again. In some sense, the frequency of A5 is 880Hz, but it is also vibrating at 440Hz. The converse is not true: the frequency of A4 is not 880Hz, because after 1/880th of a second from being plucked, the A4 string will not be back where it started. The "frequency" that we assign to a named note is related to the shortest time it takes for the vibration to get back to its original position. So, if it takes you 50s to run a lap around the track, you can run two laps in 100s and three laps in 150s. In fact, any multiple of 50s will see you right back at the start (==finish) line. A5 can run a lap exactly twice as fast as A4, so every time A4 makes a lap, she sees A5 at the start line. A5 only sees A4 every other time.
Here is a simplified illustration. In the following picture, the x-axis is time, and the y-axis is the extent of your guitar string up vs down after you pluck it (or the pressure of the air, high vs low at some point near your ear, or the extent of your eardrum, in vs out).
The 0.01 on the x-axis is in seconds. That's right, remember that our string is going back and forth at 440Hz - that's 440 times per second! For reference, low-E on your guitar is E2 at about 82Hz. Pluck your low E string and see if you can count the number of times the string goes back and forth in one second. :)
A5 is higher pitched, meaning that the string is vibrating faster: 880 times per second.
If we plot them together, we see that every other time our A5 string is at the top of its vibration, our A4 string is also at the top of its vibration! In other words, both A4 and A5 vibrate at 440Hz (but A5 also vibrates at 880Hz, and A4 does not). There's nothing particularly special about the top: if we plucked them at slightly different times, then perhaps our A4 string would be at the top every other time A5 is at the middle (different "phase"). It's happening so fast that we don't "hear" the phase, but we do hear the alignment. The important aspect here is the lining up "every other time". That is what musicians mean by calling both of these notes "A". The fact that its "every other time" and not "every time" is the difference between A4 and A5.
