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!