TITLE: What is the difference between a 1-form and a co-vector?
QUESTION [7 upvotes]: My understanding has been that co-vectors are defined on a (pseudo-) Riemannian manifold as dual vectors with respect to the given metric. However, 1-forms do not need the notion of a metric and have nothing to do with vectors except that one feeds on the other.
However, while reading this answer on the local representation of a tautological form (which is defined on a general manifold irrespective of the presence of a metric structure), I cannot seem to comprehend the difference between a 1-form and a co-vector, which difference seems important to the answer.
Please let me know what I am missing. If possible, please also answer the linked question bearing in mind my lack of comprehension of this difference. Thank you.

REPLY [7 votes]: Evidently (see comment from Nanashi No Gombe under question) this was only a matter of mismatched naming conventions, but here is a TL;DR answer for future readers, which (in my experience) matches the way most everyone uses these terms:
A co-vector is to a $1$-form as a vector is to a vector field.
That is, if you have a $1$-form $\alpha$ on some manifold $M$ and you pick any $x \in M$, the value of $\alpha$ at $x$ is a co-vector.
If you think of a co-vector $\alpha|_x$ as a map $T_x M \rightarrow \mathbb{R}$, then you can extend this to thinking of a $1$-form as a map $\alpha: \Gamma (TM) \rightarrow \mathcal{C}^{\infty}(X)$.