The relation between the definition of quantum group and correpsonding lie algebra is discribed here. Are there some similar relation between $U_q(\widehat{sl_2})$ and $\widehat{sl_2}$? There are two definitions of $U_q(\widehat{sl_2})$.
The following is Jimbo presentation. The quantum affine algebra $U_q(\widehat{\mathfrak{sl}_2})$ is an associated algebra generated by $e_0^{\pm}, e_1^{\pm}, K_0^{\pm 1}, K_1^{\pm 1}$ subject to the relations: \begin{align} & K_0 K_1 = K_1 K_0, \\ & K_i K_i^{-1} = K_i^{-1} K_i = 1, \ i=0,1, \\ & K_i e_i^{\pm} = q^{\pm 2} e_i^{\pm} K_i, \ i=0,1, \\ & K_i e_j^{\pm} = q^{\mp 2}e_j^{\pm} K_i, \\ & e_0^{\pm} e_1^{\mp} = e_1^{\mp} e_0^{\pm} \\ & e_i^{+} e_i^{-} - e_i^{-}e_i^{+} = \frac{K_i-K_i^{-1}}{q-q^{-1}}, \ i=0,1, \end{align} and quantized Serre relations: \begin{align} (e_i^{\pm})^3 e_j^{\pm} - [3] (e_i^{\pm})^2 e_j^{\pm}e_i^{\pm} + [3]e_i^{\pm} e_j^{\pm} (e_i^{\pm})^{2} -e_j^{\pm}(e_i^{\pm})^3 = 0, \ i \neq j. \end{align}
The following is Drinfeld presentation. The quantum affine algebra $U_q(\widehat{\mathfrak{sl}_2})$ is an associated algebra generated by $x_m^{\pm}, h_r, K^{\pm 1}, c^{\pm 1}$, $m \in \mathbb{Z}, r \in \mathbb{Z}-\{0\}$, subject to the relations: \begin{align} & c^{\pm 1} \text{ are in the center}, \\ & K K^{-1} = K^{-1} K = 1, \\ & c c^{-1} = c^{-1} c = 1, \\ & [K, h_r] = 0, \\ & K x^{\pm} = q^{\pm 2} x_i^{\pm} K, \\ & [h_k, x_l^{\pm}] = \pm \frac{1}{k}[2k]c^{\mp |k|} x_{k+l}^{\pm}, \\ & x_{k+l}^{\pm} x_l^{\pm} - q^{\pm 2}x_l^{\pm} x_{k+l}^{\pm} = q^{\pm 2} x_k^{\pm} x_{l+1}^{\pm} - x_{l+1}^{\pm} x_k^{\pm}, \\ & [h_k, h_l] = \delta_{k,-l} \frac{1}{k} [2k] \frac{c^k-c^{-k}}{q-q^{-1}}, \\ & [x_k^+, x_l^-] = \frac{1}{q-q^{-1}} [c^{k-l}\psi_{k+l}-\phi_{k+l}], \end{align} where $\psi_k$, $\phi_k$ are given by: \begin{align} & \sum_{m=0}^{\infty} \psi_m z^m = K \exp\left( (q-q^{-1}) \sum_{s=1}^{\infty} h_s z^s \right), \\ & \sum_{m=0}^{\infty} \phi_{-m}z^{-m} = K^{-1} \exp\left( -(q-q^{-1}) \sum_{s=1}^{\infty} h_{-s} z^{-s} \right), \end{align} and $\psi_k=0$, $\phi_{-k}=0$, $k < 0$.
The definition of $\widehat{sl_2}$ is as follows.
Thank you very much.
First about the on the relation between $\widehat{\mathfrak{sl}_2}$ and Jimbo's presentation: For this, one needs to understand that $\widehat{\mathfrak{sl}_2}$ has the generalized Cartan matrix $$ A=(a_{ij})=\pmatrix{2&-2\\ -2& 2}. $$ To see this in more detail, you could for example look at Lecture Notes by Iain Gordon. Given a generalized Cartan matrix, Jimbo's presentation makes sense too in exactly the same way. There are now two generators in the positive part, and two in the negative part. There are denoted $e_0^+, e_1^+$, respectively ${e_0}^-, {e_1}^-$. The relations are obtained in the same way as for finite Cartan data by plugging in $a_{ij}$ in the bosonization and Serre relations.
Now, Drinfeld's presentation is different and one has to prove the two are isomorphic. For this, we can look at a paper by Beck. Note that 1.3 in 2 is the first presentation you mention. The theorem that this is isomorphic to Drinfeld's presentation is given in 4.7.
The paper 2 even gives further information about the relation to the affine Lie algebra, which is recovered by specialization of $q$ to 1 in Section 2.