It is known that the group $C(\Bbb R)$ has at most two connected components, and the connected component of the identity is isomorphic to $U(1)$ as a topological group (trivially) and $C(\Bbb Q)$ is isomorphic to $\Bbb Z^r\times E(\Bbb Q)$, where all possible $E(\Bbb Q)$ are known.
Are there similar results for other fields? I am especially interested in the case K is a number field, or the p-adic numbers (both $\Bbb Z_p$ and $\Bbb Q_p$)
By Mordell-Weil, for any number field $K$ we have
$$C(K)=\mathbb{Z}^r \times E(K)_{\mathrm{tors}}$$
As you mention, Mazur showed all the possible options for $E(\mathbb{Q})_{\mathrm{tors}}$ in his famous 1977 paper.
The only other $K$ for which we have a torsion theorem are the quadratic fields. This is the result of a long series of papers by Kamienny, Kenku and Momose from 1982 to 1992. In particular,
$$\begin{equation} E(K)_{\mathrm{tors}} \cong \begin{cases} \mathbb{Z}/m\mathbb{Z} & \text{for} 1\leq m\leq 18, m\neq 17 \\ \mathbb{Z}/2\mathbb{Z} \oplus\mathbb{Z}/2m\mathbb{Z} & \text{for}\,\, 1\leq m\leq 6 \\ \mathbb{Z}/3\mathbb{Z} \oplus\mathbb{Z}/3m\mathbb{Z} & \text{for}\,\, 1\leq m\leq 2 \\ \mathbb{Z}/4\mathbb{Z}\oplus\mathbb{Z}/4\mathbb{Z} \end{cases} \end{equation}$$
For number fields of degree $>2$ the problem is open, although quite a lot is known for degree $\leq 5$. A (probably not up-to-date) survey on partial results:
For $K$ a local field, I think it is an old result of Mattuck that $C(K)$ is a topologically finitely generated abelian profinite group. Not sure if anything else is known.
The structure of $E(K)$ for $K$ a complete local field, say a finite extension of $\mathbb Q_p$ or $\mathbb C_p$, is quite standard. Let $E_0(K)$ denote the set of points with good reduction. Then there are exact sequences $$ 0\to E_0(K)\to E(K) \to \Phi \to 0 $$ and $$ 0 \to E_1(K) \to E_0(K) \to \tilde E^{\text{ns}}_p(\mathbb F) \to 0 .$$ Here $\Phi$ is a finite group, the group of components on the Neron model. It is either cyclic of order $\text{ord}_v(\text{Disc}(E/K))$, or it is one of the groups $1,C_2,C_3,C_4,C_2\times C_2$.
$\tilde E^{\text{ns}}_p(\mathbb F)$ is the group of non-singular points defined over the residue field $\mathbb F$. In the case of singular reduction, it is either the additive group $\mathbb F^+$, or a (possibly) twisted form of the multiplicative group $\mathbb F^*$.
Finally, $E_1(K)$ is isomorphic to the group of points of a formal group. In particular, if $K$ is a finite extension of $\mathbb Q_p$, it has a subgroup of finite index that isomorphic to the additive group of the ring of integers of $K$.
The answer to one possible interpretation of the title question -- vary over all elliptic curves over all fields and ask which groups arise -- is given in this paper.
With regard to the structure of Mordell-Weil groups of elliptic curves over local fields, I think you will find that $\S$5.1 of this joint paper with Allan Lacy relevant. (The title is "Mordell-Weil Groups of Abelian Varieties Over Local Fields".) The case of characteristic $0$ is a slightly more general and less precise variant of Joe Silverman's answer, but things work a bit differently in positive characteristic.
I assume in the question that $C = E$ is an elliptic curve.
First your claim that $E(\mathbb{R}) = U(1)$ is false; I mean $E(\mathbb{R})$ can be disconnected. The correct result is that $E(\mathbb{R})$ has at most two connected components, and the connected component of the identity is isomorphic to $U(1)$ as a topological group.
As for points over other local fields. Authors usually consider the set of points which have non-singular reduction modulo $p$, as these are easier to understand that the set of all points. Some general results on the structure of these groups can be found in Chapter VII of Silverman's book "arithmetic of elliptic curves".
There are quite a few results in the literature describing these groups in special cases and there are many possible cases which can arise, depending on the type of bad reduction. For a recent result on this topic see the following:
