TITLE: Different definitions of a "one-form"
QUESTION [24 upvotes]: I started self-studying some differential geometry while using several different sources, but I'm confused about the notion of a one-form and how different places define it differently. Here are some of the definitions I've seen:

A covector field, i.e given a point $p \in M$, a one-form $\theta$ sends $p$ to $\theta(p) \in T^{*}_p M$, the cotangent space at $p$.
A function from the space of vector fields on a manifold, to $C^{\infty}(M)$.
A linear functional on a vector space.

Can anyone expand on the relation between these definitions and which one I should be think about when someone mentions "one-form"? My instinct is to use the first one, since that way, I can simply think of it as being something "dual" to vector fields.
Edit: 

Yet another place defines it as simply being a member of the cotangent space. What is all this madness?

REPLY [8 votes]: "What is all this madness?"

Welcome to differential geometry, where the notation's made up and the signs don't matter (usually).
As others have said, Definition (1) is the standard one.  A differential $1$-form is a smooth map $\theta\colon M \to T^*M$ such that $\theta(p) \in T_p^*M$ for each $p \in M$.
This is equivalent to Definition (2).  Why?  Well:
Let $\theta\colon M \to T_p^*M$ be a differential $1$-form in the sense of Definition (1), so for each $p \in M$, we have $\theta(p) \in T_p^*M$.  This means that at each $p \in M$, we have a linear map $\theta(p)\colon T_pM \to \mathbb{R}$.  That is, $\theta(p)$ is a function that inputs vectors $X_p \in T_pM$ and outputs real numbers $\theta(p)(X_p) \in \mathbb{R}$.
Thus, given a single vector $X_p \in T_pM$, we associate the scalar $\theta(p)(X_p) \in \mathbb{R}$.  Given a vector field $X$, we associate the scalar field (function) $p \mapsto \theta(p)(X_p)$.  In this way, we can regard $\theta$ as a map $\widehat{\theta}\colon \{\text{vector fields}\} \to C^\infty(M)$, which is Definition (2).
Conversely, given a map $\widehat{\theta}\colon \{\text{vector fields}\} \to C^\infty(M)$, one can define a differential $1$-form in the sense of Definition (1) by $\theta\colon M \to T^*M$ via $p \mapsto [X_p \mapsto \widehat{\theta}(X)(p)]$.  That is, $\theta(p) \in T_p^*M$ is the linear map $\theta(p)\colon T_pM \to \mathbb{R}$ defined by $\theta(p)(X_p) = \widehat{\theta}(X)(p)$.

What about Definition (3)? Now for some reason, many linear algebra books have taken to calling any functional on an arbitrary vector space a "$1$-form."  This is clearly not the same thing as a differential $1$-form as given by Definition (1).  The connection is that to any differential $1$-form $\theta\colon M \to T^*M$, the maps $\theta(p) \colon T_pM \to \mathbb{R}$ are linear functionals (hence "$1$-forms").
Finally, as you notice in the Edit, some authors describe the images $\theta(p) \in T_p^*M$ as being the $1$-forms.  This is in keeping with the linear algebra terminology described in the previous paragraph.  However, I personally don't like this conflation very much and avoid it whenever possible.
Also (Edit): What's really going on in these last two paragraphs is this: just as a vector field is a field of vectors, and a function is a field of scalars, so is a differential 1-form a field of linear functionals on $T_pM$ (aka "$1$-forms" on $T_pM$).  It might sound odd to say, but many times it causes little harm to conflate an object (like a vector) with a field of objects (like a vector field).  That's what's happening here.