I'm having understanding what it means to be a differential form on a differential manifold. I think I now understand differential forms on Euclidean space but I don't understand how this transfers to manifolds. I have that the definition of a k-form on a manifold is:
Let $M$ be a manifold with charts $\gamma_i: U_i\rightarrow V_i$ where $U_i\subseteq \mathbb{R}^n$. A $k$-form, $\alpha$, on $M$ is a collection of $k$-forms $\alpha_i$, on $U_i$ satisfying: $\alpha_j=(\gamma_i^{-1}\circ \gamma_j)^*(\alpha_i)$
Could someone please give an example of a differential form on a manifold, perhaps with the circle or sphere as the manifold.
Your definition of $k$-form is very poor. It seems the old definition using change of coordinates, like in Physics. If you understand vector fields and exterior algebra you have all what you need to state the usual definition by yourself.
Certanly, a differential $k$-form is a (smooth or $C^k$) map which assigns to each point of the manifold $p$ a $k$-form $\alpha_p$ lying in $\Lambda^k(T_p^*M)$, where $T_p^*M$ denotes the dual of the tangent space at $p$. As you can see, the background of the definition is the same as in vector fields: the manifold parametrics a family of vectors.
If you are familiarized with the notion of fibre bundles, then a differential $k$-form $\alpha$ is an element of the vector bundle
$$ \Omega^k(M) = \coprod_{p\in M} \Lambda^k(T_p^*M) .$$
EDIT:
Some bibligraphy:
-Chapter 2 of Warner Foundations of differential geometry and Lie groups (I don't like it very much.)
-Harley Flanders, Differential forms with applications, Dover 1989.
-John Lee, Introduction to smooth manifolds 2nd ed, Springer. Chapter 12 is about tensors, which can help you very well. You can skip the first part about algebraic preliminaries and go directly to page 316. Anyway, chap. 14 is about differential forms (good book and modern aproach).
-Many others. I also like Manifolds and differential geometry, from Jeffrey M. Lee, AMS graduate studies in Mathematcs vol 107. But I'm sure there are many other books which I don't know and are very very good.