I am not sure it makes sense to ask "what is the p-adic local Langlands conjecture for $\mathrm{GL}_1$". Nobody has succeeded in even formulating a reasonable candidate for a p-adic LLC for $\mathrm{GL}_n$, so you can't just plug $n = 1$ in and see what it says. Morally, the conjecture should be something like this:
- for $K$ a p-adic field, there is a bijection between continuous representations $\mathrm{Gal}(\overline{K} / K) \to \mathrm{GL}_n(E)$, $E$ a p-adic field, and continuous admissible unitary p-adic $E$-Banach-space representations of $\mathrm{GL}_n(K)$, satisfying [some properties].
The problem is: what properties? Saying "there is a bijection" is just claiming that the sets have the same cardinality, which is trivial, and clearly not what is intended here (Serre makes exactly this point in his lecture How To Write Mathematics Badly, which is on YouTube). So one needs to spell out what the properties are, and this is hard to do.
In this $\mathrm{GL}_1$ setting you can cheat, because there is already a candidate for this bijection (any continuous unitary character $K^\times \to E^\times$ extends uniquely to the profinite completion, and $\widehat{K^\times}$ is the abelianization of $\mathrm{Gal}(\bar{K} / K)$ by class field theory). But one doesn't know how to uniquely pin down what the question is for $\mathrm{GL}_n$, which is one reason why it's hard to answer!
Even for $\mathrm{GL}_2 / \mathbf{Q}_p$, Colmez has constructed a bijection between p-adic Banach reps of $\mathrm{GL}_2(\mathbf{Q}_p)$ and 2-dimensional Galois reps, which has a long list of very nice properties; but as far as I'm aware there's not a theorem which says "Colmez's correspondence is the unique correspondence with properties X, Y and Z".
EDIT. Let me add some more about the global side. Here it's even less clear what the question is. For $\mathrm{GL}_n$ over a number field $K$, the statement should probably be that there should be a bijection between: $n$-dimensional $p$-adic representation of $\mathrm{Gal}(\overline{K} / K)$ which are continuous and unramified almost everywhere; and "p-adic automorphic representations of $\mathrm{GL}_n$". But it's not at all clear what the definition of the latter class of objects should be for $n \ge 2$.
The eigenvariety for $\mathrm{GL}_1 / K$, $K$ a number field, is rather well understood. I'll just summarize Buzzard's construction briefly. Choose some open compact subgroup $U$ of $\left(\mathbf{A}_K^{(p\infty)}\right)^\times$ (the finite adeles away from $p$). Let $K_\infty^\circ$ be the totally positive elements of $(K \otimes \mathbf{R})^\times$. Then the quotient
$$ G(U) = \mathbf{A}_K^\times / \overline{ U \cdot K_\infty^\circ \cdot K^\times } $$
is a pretty friendly abelian group; it has a subgroup of finite index isomorphic to $\mathbf{Z}_p^n$ for some integer $1 \le n \le [K : \mathbf{Q}]$. Moreover, Leopoldt's conjecture is very easily seen to be equivalent to determining $n$ (it says that $n = 1 + r_2$, where $r_2$ is the number of complex places of $K$). Since $G(U)$ is such a friendly group, it's easy to see that $p$-adic characters of $G(U)$ naturally biject with points of a rigid space $\mathcal{E}(U)$ and that's your eigenvariety.
Now, the points of $\mathcal{E}(U)$ can be viewed as continuous $p$-adic characters of $\mathbf{A}_K^\times / K^\times$ with prime-to-$p$ ramification bounded by $U$, or (via class field theory) as 1-dimensional Galois representations, again with prime-to-$p$ ramification bounded by $U$. That gives a bijection between these classes of objects, to which it seems natural to give the name "p-adic Langlands for $GL_1$". So there's the relation between the objects in Chandan's question:
- p-adic Langlands for $GL_1$ is a bijection between two classes of objects, "automorphic" and "Galois", which is an easy consequence of class field theory;
- both of these classes of objects are naturally parametrised by a $p$-adic rigid space, which we call the eigenvariety;
- Leopoldt's conjecture is a statement about the dimension of the eigenvariety, which involves the same objects as $p$-adic Langlands but is logically independent of it.
As soon as you step away from $\mathrm{GL}_1$, though, eigenvarieties get less central to the $p$-adic Langlands picture (although they're still clearly very important). The issue now is that eigenvarieties are expected to parametrise Galois representations which are fairly "degenerate" locally at $p$: technically they should be trianguline, a notion introduced by Colmez. For $\mathrm{GL}_1$ everything is (vacuously) trianguline, but this is further and further away from being true as $n$ grows; e.g. a Galois representation with finite image is only trianguline if it's a direct sum of characters. Trianguline representations should correspond under p-adic Langlands to representations of $\mathrm{GL}_n$ induced from the Borel subgroup -- so in a sense these are the easiest ones!
