Which representation of the Lorentz group is the analog of an $SU(2)$ triplet or $SU(2)$ $n$-plet? A contravariant vector $v^\mu$ transform as the fundamental representation of the Lorentz group as $$v^\mu=\Lambda^\mu_{~~~\nu}v^\nu$$ which is tensor of rank-one. A rank-two tensor transforms as $$t^{\mu\nu}=\Lambda^\mu_{~~~\rho}\Lambda^\nu_{~~~\sigma}t^{\rho\sigma}.$$
Similarly, an $SU(2)$ doublet is a set of two fields $\psi^i$ $(i=1,2)$ which transforms under $SU(2)$ group as $$\psi^i=U^i_{~j}\psi^i.$$ This is the fundamental representation of $SU(2)$ group. 
Therefore, the doublets $\psi^i$'s for the $SU(2)$ group are analogs of $v^\mu$ for the Lorentz group in the sense that both are fundamental representations of the respective groups. Compare first equation with third. 
Now my question is, which representations of the Lorentz group are the analogs of an $SU(2)$ triplet $\phi^i$ ($i=1,2,3$) which is a set of three fields or any $n$-plet $\phi^i$ ($i=1,2,3,..n$) which a set of $n$ fields? I hope the question is clear. Please help me with an answer as less technical as possible.
 A: *

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*The double-cover of the 3D rotation group $SO(3;\mathbb{R})$ is isomorphic to $SU(2)$. 

*The double-cover of the restricted Lorentz group $SO^+(1,3;\mathbb{R})$ is isomorphic to the complex Lie group $SL(2,\mathbb{C})$. 
The former pair are subgroups of the latter pair, respectively, cf. the diagram below.
Returning to OP's question, the $n$-dimensional irrep of $SL(2,\mathbb{C})$ restricts to the $n$-dimensional irrep of $SU(2)$, which is the $n$-dimensional projective  irrep of $SO(3;\mathbb{R})$. 
$$ \begin{array}{ccccc} SU(2) & \subseteq & SL(2,\mathbb{C})& \subseteq & SL(2,\mathbb{C})_L\times SL(2,\mathbb{C})_R \cr
 \pi\downarrow && \pi\downarrow && \pi\downarrow\cr
 SO(3;\mathbb{R}) & \subseteq & SO^+(1,3;\mathbb{R}) & \subseteq & SO(1,3;\mathbb{C})\end{array}$$


*Similarly, the double-cover of the complexified proper Lorentz group $SO(1,3;\mathbb{C})$ is isomorphic to $SL(2,\mathbb{C})_L\times SL(2,\mathbb{C})_R$, which has $SU(2)_L\times SU(2)_R$ as a subgroup. Again an irrep $(n_L,n_R)$ of the bigger group restricts to a corresponding irrep $(n_L,n_R)$ of the subgroup. 

*See e.g. this and this related Phys.SE posts for more details.
A: The way you are seeking analogs, you can only find analogs for certain specifically defined representations. For example, as you do, you can say that the fundamental of $SU(2)$ is analogous to the fundamental of $SO(1,3)$ as in that they are both fundamentals of the respective groups. Similarly, you can say that the adjoint of $SU(2)$ is analogous to the adjoint of $SO(1,3)$. But, this is almost a word-game. There is no actual correspondence between representations compared in such a manner. To wit, the "analogous" representations are not even of the same dimension.
A more useful way to go about comparing the representations of $SU(2)$ and $SO(1,3)$ is via realizing that $\mathfrak{su}(2)\times\mathfrak{su}(2)\cong\mathfrak{so}(1,3)$. So, a particular pair of representations of the two $SU(2)$s corresponds to some particular representation of $SO(1,3)$. For example, the $\text{Spin }\frac{1}{2}\times\text{Spin }0$ representation (i.e., the $\text{Doublet}\times\text{Singlet}$ representation) of $SU(2)\times SU(2)$, commonly denoted as the $(\frac{1}{2},0)$ representation, corresponds to the left-handed (Weyl) spinor representation of $SO(1,3)$. Similarly, the $(\frac{1}{2},0)\oplus(0,\frac{1}{2})$ representation of $SU(2)\times SU(2)$ corresponds to the bispinor Dirac representation of $SO(1,3)$. The fundamental representation of $SO(1,3)$, i.e., the vector representation, corresponds to the $(\frac{1}{2},\frac{1}{2})$ representation of $SU(2)\times SU(2)$. See: https://en.wikipedia.org/wiki/Representation_theory_of_the_Lorentz_group
