What is the first, second etc mode? I cannot find online explanations. Is it the shape of vibration? Does a thing have more than one natural frequencies (first, second, etc) and it vibrates with different modes in these frequencies, named 1st, 2nd etc modes? thanks!

Modes of vibration are particularly, though by no means exclusively, associated with musical instruments. It is the shape of vibration, and most musical instrument have more one mode of vibration, of they would be fairly limited in their musical range. Compare the sounds of a violin (with 4 to 7 strings) with a musical triangle, which only emits one note.

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The first 3 modes of vibration of a guitar string.

For a more extreme example of the various vibration modes possible, here are some computer generated modes from a drumhead.

Images and Extracts from Modes of Vibration

Drumhead Vibrations

When you pluck a stretched string, you always hear a sound with a definite musical pitch. By altering the length, tension or weight of the string, all familiar to musicians, you can alter this pitch. Strings and stretched drumheads are all suitable for producing a variety of vibrations, so they make musical instruments with a wide range of sounds possible. If instead you used a brick, or a frying pan, there is very little scope for musicical variety, as their vibration modes are limited.

The simplest mathematical description of the vibration of a stretched string reveals a pattern in the set of resonance frequencies. Once the lowest (or fundamental) frequency has been fixed by choosing the weight, tension and length of the string, then all the other frequencies are whole-number multiples: if the first is f, then the second is 2f, the third 3f and the nth is nf . The frequencies are called the natural frequencies or overtones, and this simple numerical pattern relating them is called a harmonic series: so a stretched string has natural frequencies which are harmonic.

Here is another example, but not musical, of modes of vibration

Galloping Gertie Movie Tacoma Narrows Bridge Collapse "Gallopin' Gertie" - YouTube

Tacoma Narrows Bridge Collapse

The failure of the bridge occurred when a never-before-seen twisting mode occurred, from winds at a mild 40 miles per hour (64 km/h). This is a so-called torsional vibration mode (which is different from the transversal or longitudinal vibration mode), whereby when the left side of the roadway went down, the right side would rise, and vice versa, with the center line of the road remaining still. Specifically, it was the "second" torsional mode, in which the midpoint of the bridge remained motionless while the two halves of the bridge twisted in opposite directions. Two men proved this point by walking along the center line, unaffected by the flapping of the roadway rising and falling to each side. This vibration was caused by aeroelastic fluttering.

Usually an object can vibrate at different frequencies. There is a lowest frequency, the ground mode, but higher frequencies are possible. The details depend on the shape and materiel properties of the vibrating body. In the most simple case the higher frequencies are multiples of the base frequency, in which case they are also called harmonics. The common case, though, is that there are much more frequencies. A simple example is a guitar string already providing a rather complex spectrum, i.e. mixture of frequencies. In this simple "1D" case the the possible frequencies are given by the possible nodes on the string. Imagine a string with length $L$ then there is a frequency corresponding to a movement with just two nodes at the end, the fundamental mode. The next higher frequency is given by one additional node in the middle, then 2 etc. with more nodes, the string has to bend more, you store more elastic energy and the force to drive it back to equilibrium becomes stronger. The string moves faster and sound is higher. In higher dimensions it is more complicated, but in principle the number of nodes increases with increasing energy and frequency, while counting or naming modes might not be unique any more. Also note that the shape and mechanical properties uniquely define the spectrum, i.e. the possible frequencies at which a body can vibrate, but knowing the spectrum does not necessarily lets you calculate the shape

Long story short: Yes, a body vibrates at different frequencies, the more nodes the vibration, the higher the frequency. Modes are numbered with increasing frequency/energy (if unique and therefore possible)

Under suitable conditions the solution of a wave equation can be written in the form: $$ f(t, \mathbf{x}) = \sum_{\mathbf{k}}a(\mathbf{k})\,\textrm{e}^{i\left(\omega(\mathbf{k}) t - \mathbf{k}\cdot\mathbf{x}\right)} $$ The $\mathbf{k}^{th}$ term $a(\mathbf{k})$ is referred to as the $\mathbf{k}^{th}$ mode, with corresponding frequency $\omega(\mathbf{k})$. The subset of values $(k_x, k_y, k_z)$ to sum over is given by exploiting the boundary conditions of the wave equations.

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