$p\ \text dq$ is the "tautological" one-form?

The name seems appropriate if consider that it probably comes from the case when the manifold is the cotangent bundle of a manifold. Then a point on $T^*M$ is a pair $(x,\alpha)$, where $x$ is a point on $M$ and $\alpha$ a one form. The definition of the tautological one form is: the value of the form at a point $(x,\alpha)$ on a tangent vector is obtained by projecting the vector to a tangent vector on $M$ and evaluating (at $x$) the form $\alpha$. In other words for $v\in T(T^*M)$ the value is $\theta_{(x,\alpha)}(v):=\alpha_x(d\pi(v))$, where $\pi : T^*M \rightarrow M$ is the projection.

In a way it is pretty tautological.

I) On a general symplectic manifold $({\cal M},\omega)$ (typically called phase space by physicists), one can locally choose a symplectic potential $\theta\in \Gamma(T^{*}{\cal M}|_{\cal U})$, which is a one-form such that

$$\tag{1} \mathrm{d}\theta~=~\omega,$$

cf. Poincare Lemma. Here ${\cal U}\subseteq {\cal M}$ denotes a local neighborhood.

Note that the symplectic potential $\theta$ is never unique (or 'canonical') in the sense that

$$\tag{2} \theta^{\prime}~=~\theta+\mathrm{d}F$$

would also be a symplectic potential, if $F$ is a zero-form (aka. a function).

For a general symplectic manifold $({\cal M},\omega)$ there may not exist a globally defined symplectic potential $\theta$.

Darboux' theorem states that any $2n$-dimensional symplectic manifold $({\cal M},\omega)$ is locally isomorphic to the cotangent bundle $T^*(\mathbb{R}^n)$ equipped with the canonical symplectic two-form.

II) Consider next the special case where the symplectic manifold ${\cal M}=T^{*}M$ happens to be a cotangent bundle

$$\tag{3} {\cal M}~=~T^{*}M~\stackrel{\pi}{\longrightarrow}~ M $$

equipped with the canonical symplectic two-form $\omega$, which in local coordinates reads

$$\tag{4} \omega|_{\pi^{-1}(U)} ~=~\sum_{i=1}^n\mathrm{d}p_i\wedge \mathrm{d}q^i.$$

Here $U\subseteq M$ denotes a local neighborhood of the base manifold $M$. (The base manifold $M$ is typically called the configuration space by physicists.) Moreover, $q^i$ are local coordinates on the base manifold $M$, and $p_i$ are local coordinates of the cotangent fibers.

Then there always exists a globally defined symplectic potential $\theta\in \Gamma(T^{*}{\cal M})$ that in local coordinates reads

$$\tag{5} \theta|_{\pi^{-1}(U)}~=~\sum_{i=1}^n p_i ~\mathrm{d}q^i.$$

Since the globally defined one-form (5) comes for free on a cotangent bundle ${\cal M}=T^{*}M$ for any manifold $M$, it is 'tautological' in that sense. But wait, there there is more: It can be defined in a manifestly coordinate-independent way, cf. Wikipedia & MBN's answer.