Timbre comes from old French meaning "bell" or "drum," which later came to mean "sound" or "tone" and is pronounced \TAM-ber\ with a more French influenced second syllable \TAM-bruh\. Timbre initially meant a drum and came to French from Greek word τύμπανον (týmpanon) meaning “kettledrum”.
Sound has 2 (among some others) really basic characteristics: Pitch (frequency) and Loudness (amplitude).
Pitch is purely psychological construct which relates to the actual frequency of a particular sound. Loudness too is a purely psychological construct that relates to how much energy a source of sound creates: how much air it displaces when the sound is made. When I say psychological construct what I mean is they only exist in our minds and do not exist in nature on their own. Our minds bring pitch and loudness in to reality.
The sounds you just played specifically middle C4 is a musical note with a frequency of 261.63Hz. C4 has a scientific frequency of 256Hz which uses mathematical ways to measure the frequency while the musical pitch of 261.63Hz is used my musicians for tuning their instruments. All the 4 instruments piano, trumpet, violin, and flute are playing the same frequency of 261.63Hz but yet they sound different. This is timbre, the unique "sound colour" of an instrument. When they play the note the sound vibrations are oscillating at approximately 262Hz and yet they sound different.
Why?
Let's find out

Nearly all objects when hit, pluck, strummed or somehow disturbed vibrate. The frequency at which the objects vibrate is called its natural frequency. Let's take an example of a string instrument like a guitar. When you pluck a guitar string it vibrates along it full length at a frequency called the fundamental frequency also called as the first harmonic. So in case of a middle C4 the string vibrates approximately 262 times per second (Hertz). This vibration at the fundamental frequency gives the guitar string it's base pitch which we hear. The frequency is determined by various factors and each of the factors affect either the wavelength of the object since frequency = speed / wavelength. For a guitar with 6 strings the factors include the linear density of the strings (wider strings are more dense), different tension of the strings, and the different length of the strings. The speed at which a wave travels through a string depends on the properties of the medium -- tension and linear density. Changes in these properties affect the natural frequency of the string.

So when a guitarist plucks a string of a guitar the plucking transfers energy to the string and it starts to vibrate. The disturbance travels in both the directions. Since the string is fixed by the frets on one side and the bridge on the other the string is unable to move and once the disturbance travels from one end to the other and is reflected back to travel in the opposite direction. When the disturbance travels back it interferes with the forward moving disturbance (wave). These points are called nodes -- the points of no displacement. In between two nodes there has to be an antinode. The fundamental frequency of the string is associated with one antinode between the two nodes. The string vibrates in such a way that the displacement is zero at both ends (nodes), and there is a maximum displacement at the center of the string (an antinode). The string vibrates with the lowest frequency possible, corresponding to the longest wavelength that fits within the string's length.

In comes resonance. The string has a natural set of frequencies at which it vibrates most strongly. These natural frequencies are the fundamental and its harmonics. Every string has specific frequencies at which it naturally vibrates based on its:

1. Length: The longer the string, the lower the fundamental frequency. f = 1/length
2. Tension: The tighter the string, the higher the frequencies. F = √T
3. Density: Frequency decreases as density of the material increases. F = 1/√μ

These are called the string's resonant frequencies. When you pluck the string, it will vibrate at one or more of these resonant frequencies, and this is where resonance occurs. The string has natural resonant frequencies that it “prefers” to vibrate at. These are not just at the fundamental frequency, but at integer multiples of that frequency. These higher frequencies are called harmonics or overtones. Timbre is a consequence of the overtones. The string can also vibrate in more complex modes by dividing the string into sections. Each section vibrates independently at a higher frequency. These modes of vibration occur at specific frequencies where the string fits an integer number of half-wavelengths along its length. An overtone is any frequency higher than the fundamental in the harmonic series. The fundamental and the overtones are called partials. Harmonic partials are the frequency which are integer multiple of the fundamental including the fundamental frequency where the relationship is by 1 times itself.

Overtones:

• any frequency component above the fundamental
• can be harmonic or inharmonic
• includes all the frequencies generated by the vibrating system

Harmonics:

• all the frequencies including the fundamental
• frequencies that are integer multiple of the fundamental
• follow a particular mathematical relationship eg. 2x, 3x, 4x of the fundamental

All harmonics are overtones but not all overtones are harmonics

So when a guitar string is plucked it's vibrating at a fundamental frequency and also at its resonating frequencies giving rise to harmonics and a rich set of frequencies.

For a string fixed at both ends:
Fundamental Frequency (1^st harmonic): The standing wave has 1 antinode in the middle and 2 nodes at the ends. The wavelength is twice the length of the string. Relation: λ₁ = 2L
2^nd harmonic: The standing wave has 2 antinodes and 3 nodes (nodes at both ends, one in the middle, and two between them). The wavelength is the length of the string. Relation: λ₂ = L
3^rd harmonic: The standing wave has 3 antinodes and 4 nodes The wavelength is the length of the string. Relation: λ₃ = 2L/3
n^th Harmonic: The standing wave has n antinodes and n+1 nodes. The wavelength for the nth harmonic is given by: λ_n = 2L/n

The same logic can be applied to the wind instruments. Open pipes (e.g. flutes, recorders) open at both ends, these instruments produce all harmonics (1st, 2nd, 3rd, etc.). The fundamental frequency (f₁) is supported, along with integer multiples: f₂ = 2f₁ f₃ = 3f₁ and so on. In an open pipe, both ends act as antinodes, where the air vibrates with maximum amplitude. The relationship between the wavelength (λ), frequency (f), and the length (L) of the pipe is given by: L = nλ/2 f = nv/L where:
n = Harmonic number (1st harmonic, 2nd harmonic, etc.).
v = Speed of sound in air (≈343 m/s at room temperature)

Closed pipes (e.g. clarinets, organ pipes) closed at one end, these instruments produce only odd harmonics (1st, 3rd, 5th, etc.). The fundamental frequency (f₁) is supported, and higher harmonics are odd multiples: f₃ = 3f₁ f₅ = 5f₁ and so on. In a closed pipe, the closed end acts as a node (no air movement), and the open end acts as an antinode (maximum air movement). The relationship between the wavelength (λ), frequency (f), and the length (L) is: L = nλ/4 f = nv/4L where:
n = Harmonic number (1st harmonic, 2nd harmonic, etc.).
v = Speed of sound in air
Brass instruments with valves or slides (e.g., Trumpets, Trombones) rely on the player's lips to excite the air column, producing a mix of harmonics. The harmonic series depends on the length of the tube, which is modified by valves or slides.

Different materials have different densities. Different objects make different sound partly because of density and partly due to their shape when they are struck by something. If strike an object energy from the strike causes the molecules within them to vibrate at several different frequencies at the same time determined by the shape, material, density, and size. When an object vibrates at let's say 100Hz, 200Hz, 300Hz, 400Hz, 500Hz, etc. the intensity of vibration at each of the harmonic is not the same, it varies. If you play an trumpet at 200Hz you hear many tones not just one. The other tones are multiple integers of the fundamental frequency: 400Hz, 660Hz, 880Hz, etc. These overtones have different energy levels so hear them as having different loudness. The particular pattern of the loudness of the overtones creates a sonic fingerprint for an instrument. A violin playing the same 200Hz note will have the same frequencies but different pattern of the loudness of the overtones with respect to others. The unique sound that gives trumpet its trumptiness comes from unique way how the loudness levels are distributed across the overtones. Each instrument has a tonal profile which is like its fingerprint. Trumpets have a tonal profile where there is relatively even amount of energy in odd and even harmonics both. A violin that is bowed in the center will produce mostly odd harmonics but bowing at the one third of the length emphasizes the third harmonic and its multiple: sixth, ninth, etc.
All instruments (and even human voice) have a timbral fingerprint which is different to other instruments. But to the trained ear or musicians not all trumpets sound alike nor do all pianos. Every instrument has a different overtone profile but not so different that it starts to sound like some other instrument. Master musicians can hear the subtle differences in the sound. That is timber.