One needs to carefully distinguish between the classical and the quantum mechanical treatment here. For simplicitly, I will talk about the bosonic string, but the pertinent points also carry over to the superstring.

The classical string is literally a one-dimensional object floating in some n-dimensional Lorentzian spacetime. In analogy to a relativistic particle, it traces out a worldsheet, and its motion is governed by a rule to extremize the Lorentzian area of this worldsheet, just like particles extremize the length of their worldlines between two fixed event (i.e. they follow geodesics). The action for this motion is the famous Nambu-Goto action. It is classically equivalent to the computationally much mroe tractable Polyakov action, whose equation of motion for the string coordinates $X^\mu(\sigma,\tau)$, where $\sigma$ is the spatial position along the string and $\tau$ the time along the worldsheet is just a simple wave equation: $$ \left(\partial_\tau^2 - \partial_\sigma^2\right)X^\mu = 0$$ On a closed string, we have $X^\mu(l,\tau) = X^\mu(0,\tau)$ as a boundary condition. We can then write down the general form of a solution to this wave equation: $$ X_\pm^\mu(\tau\pm\sigma) = \underbrace{\frac{x^\mu + c^\mu}{2}}_{\text{initial position}} + \underbrace{\frac{\pi\alpha'}{l}p^\mu(\tau\pm\sigma)}_{\text{center-of-mass motion}} + \underbrace{\sum_{n\in\mathbb{Z}-\{0\}} \frac{\alpha^{\pm\mu}_n}{n}\mathrm{e}^{-\frac{2\pi\mathrm{i}}{l}n(\tau\pm\sigma)}}_{\text{vibrational modes}}$$ Here, the $x$ and $c$ are for our purposes irrelevant constants, and likewise the $p$ and $\alpha'$. The relevant term is the last one: The $\frac{1}{n}$ is a normalization factor, the $\alpha_n$ a Fourier coefficient (i.e. one should think of a function $\alpha^{\pm\mu} : \mathbb{Z}\to\mathbb{R}, n\mapsto \alpha^{\pm\mu}_n$ as being the Fourier transform of $X^\mu_\pm$), and the exponential simply the $n$-th harmonic vibration on the string. Neither the derivation of this solution nor its exact mathematical form is relevant for the basic thrust of this answer.

Visualizing these vibrations is as easy as imagining a (non-streching) thin wristband flying through a room. Once in a lifetime you might get a pure harmonic vibration on it, but generally, it will just wobble about in a superposition of the many possible modes.

One needs to carefully distinguish between the classical and the quantum mechanical treatment here. For simplicity, I will talk about the bosonic string, but the pertinent points also carry over to the superstring.

The classical string is literally a one-dimensional object floating in some n-dimensional Lorentzian spacetime. In analogy to a relativistic particle, it traces out a worldsheet, and its motion is governed by a rule to extremize the Lorentzian area of this worldsheet, just like particles extremize the length of their worldlines between two fixed event (i.e. they follow geodesics). The action for this motion is the famous Nambu-Goto action. It is classically equivalent to the computationally much more tractable Polyakov action, whose equation of motion for the string coordinates $X^\mu(\sigma,\tau)$, where $\sigma$ is the spatial position along the string and $\tau$ the time along the worldsheet is just a simple wave equation: $$ \left(\partial_\tau^2 - \partial_\sigma^2\right)X^\mu = 0$$ On a closed string, we have $X^\mu(l,\tau) = X^\mu(0,\tau)$ as a boundary condition. We can then write down the general form of a solution to this wave equation: $$ X_\pm^\mu(\tau\pm\sigma) = \underbrace{\frac{x^\mu + c^\mu}{2}}_{\text{initial position}} + \underbrace{\frac{\pi\alpha'}{l}p^\mu(\tau\pm\sigma)}_{\text{center-of-mass motion}} + \underbrace{\sum_{n\in\mathbb{Z}-\{0\}} \frac{\alpha^{\pm\mu}_n}{n}\mathrm{e}^{-\frac{2\pi\mathrm{i}}{l}n(\tau\pm\sigma)}}_{\text{vibrational modes}}$$ Here, the $x$ and $c$ are for our purposes irrelevant constants, and likewise the $p$ and $\alpha'$. The relevant term is the last one: The $\frac{1}{n}$ is a normalization factor, the $\alpha_n$ a Fourier coefficient (i.e. one should think of a function $\alpha^{\pm\mu} : \mathbb{Z}\to\mathbb{R}, n\mapsto \alpha^{\pm\mu}_n$ as being the Fourier transform of $X^\mu_\pm$), and the exponential simply the $n$th harmonic vibration on the string. Neither the derivation of this solution nor its exact mathematical form is relevant for the basic thrust of this answer.

Visualizing these vibrations is as easy as imagining a (non-stretching) thin wristband flying through a room. Once in a lifetime you might get a pure harmonic vibration on it, but generally, it will just wobble about in a superposition of the many possible modes.

#The quantum string

The quantum string

The classical string is literally a one-dimensional object floating in some n-dimensional Lorentzian spacetime. In analogy to a relativistic particle, it traces out a worldsheet, and its motion is governed by a rule to extremize the Lorentzian area of this worldsheet, just like particles extremize the length of their worldlines between two fixed event (i.e. they follow geodesics). The action for this motion is the famous Nambu-Goto action. It is classically equivalent to the computationally much mroe tractable Polyakov action, whose equation of motion for the string coordinates $X^\mu(\sigma,\tau)$, where $\sigma$ is the spatial position along the string and $\tau$ the time along the worldsheet is just a simple wave equation: $$ \left(\partial_\tau^2 - \partial_\sigma^2\right)X^\mu = 0$$ On a closed string, we have $X^\mu(l,\tau) = X^\mu(0,\tau)$ as a boundary condition. We can then write down the general form of a solution to this wave equation: $$ X_\pm^\mu(\tau\pm\sigma) = \underbrace{\frac{x^\mu + c^\mu}{2}}_{\text{initial position}} + \underbrace{\frac{\pi\alpha'}{l}p^\mu(\tau\pm\sigma)}_{\text{center-of-mass motion}} + \underbrace{\sum_{n\in\mathbb{Z}-\{0\}} \frac{\alpha^{\pm\mu}_n}{n}\mathrm{e}^{-\frac{2\pi\mathrm{i}}{l}n(\tau\pm\sigma)}}_{\text{vibrational modes}}$$ As you might recall, waves can be split into a right-moving and a left-moving part, the $X_-$ is "right"-moving and the $X_+$ is "left"-moving in our convention. This is nothing but the decomposition of the string motion into the different parts of its motion. Most notably, the last term is just a sum over the Fourier modes of the string - it represents the actual, classical vibration of the string. Each $\alpha^{\pm\mu}_n$ is a unique, well-defined harmonic mode; the entirety of the modes gives a vibration which of course need not correspond to any nice harmonic vibration.

The classical string is literally a one-dimensional object floating in some n-dimensional Lorentzian spacetime. In analogy to a relativistic particle, it traces out a worldsheet, and its motion is governed by a rule to extremize the Lorentzian area of this worldsheet, just like particles extremize the length of their worldlines between two fixed event (i.e. they follow geodesics). The action for this motion is the famous Nambu-Goto action. It is classically equivalent to the computationally much mroe tractable Polyakov action, whose equation of motion for the string coordinates $X^\mu(\sigma,\tau)$, where $\sigma$ is the spatial position along the string and $\tau$ the time along the worldsheet is just a simple wave equation: $$ \left(\partial_\tau^2 - \partial_\sigma^2\right)X^\mu = 0$$ On a closed string, we have $X^\mu(l,\tau) = X^\mu(0,\tau)$ as a boundary condition. We can then write down the general form of a solution to this wave equation: $$ X_\pm^\mu(\tau\pm\sigma) = \underbrace{\frac{x^\mu + c^\mu}{2}}_{\text{initial position}} + \underbrace{\frac{\pi\alpha'}{l}p^\mu(\tau\pm\sigma)}_{\text{center-of-mass motion}} + \underbrace{\sum_{n\in\mathbb{Z}-\{0\}} \frac{\alpha^{\pm\mu}_n}{n}\mathrm{e}^{-\frac{2\pi\mathrm{i}}{l}n(\tau\pm\sigma)}}_{\text{vibrational modes}}$$ Here, the $x$ and $c$ are for our purposes irrelevant constants, and likewise the $p$ and $\alpha'$. The relevant term is the last one: The $\frac{1}{n}$ is a normalization factor, the $\alpha_n$ a Fourier coefficient (i.e. one should think of a function $\alpha^{\pm\mu} : \mathbb{Z}\to\mathbb{R}, n\mapsto \alpha^{\pm\mu}_n$ as being the Fourier transform of $X^\mu_\pm$), and the exponential simply the $n$-th harmonic vibration on the string. Neither the derivation of this solution nor its exact mathematical form is relevant for the basic thrust of this answer.

As you might recall, waves can be split into a right-moving and a left-moving part, the $X_-$ is "right"-moving and the $X_+$ is "left"-moving in our convention. This is nothing but the decomposition of the string motion into the different parts of its motion. Most notably, the last term is just a sum over the Fourier modes of the string - it represents the actual, classical vibration of the string. Each $\alpha^{\pm\mu}_n$ is a unique, well-defined harmonic mode; the entirety of the modes gives a vibration which of course need not correspond to any nice harmonic vibration.