Let $X=\{a,b,c,d\}$. Find the connected component $C(a)=\bigcup \{A \mid a \in A , A \subset X, \text{$A$ connected}\}$. 
Let $X=\{a,b,c,d\}$ and $\tau=\{\emptyset, \{a,b\}, \{a,b,c\}, \{c,d\}, X\}$. Find the connected component $C(a)=\bigcup \{A \mid a \in A  , A \subset X, \text{$A$ connected}\}$.

I'm trying to build intuition for connected components of topological space and I have trouble with determining if a subset of a space $X$ is connected.
I know that a space is connected if for some non-empty $U,V \in \tau$ we have that $U \cap V =X$ and $U \cap V = \emptyset$.
In this case I think I should consider sets $\{a\}, \{a,b\}, \{a,c\}, \{a,d\}, \{a,b,c\}, \{a,c,d\}, \{a,b,d\}, \{a,b,c,d\}$. I think these cover all the subsets of $X$ with $a$ fixed since there is $2^{n-1}$ subsets of a set with one element fixed.
But I don't know how to check whether these sets are connected? A subspace seems to be connected if it's connected in it's relative topology, but what is the relative topology of these subsets anyway here?
Are the relative topologies here $$\tau_{\{a\}}=\{\emptyset, \{a\}\} \\\tau_{\{a,b\}}=\{\emptyset, \{a,b\}\} \\\tau_{\{a,c\}}=\{\emptyset,\{a\}, \{a,c\}, \{c\}\}\\ \tau_{\{a,d\}}=\{\emptyset, \{a\}, \{a,c\}, \{c\}\} \\ \tau_{\{a,b,c\}}=\{\emptyset, \{a,b\},\{a,b,c\}, \{c\}\} \\ \tau_{\{a,c,d\}}=\{\emptyset, \{a\},\{a,c\},\{c,d\}\} \\ \tau_{\{a,b,d\}} = \{\emptyset,\{a,b\},\{d\}\}$$
and for the last set $\{a,b,c,d\}$ the topology would just be $\tau$ as it's the universal set.
 A: "I know that a space is connected if for some non-empty $U,V \in \tau$ we have that $U \cap V =X$ and $U \cap V = \emptyset$."
Attention: there's a mistake, it should be: $U,V \in \tau$ we have that $U \cup V =X$ and $U \cap V = \emptyset$
And in order to find the conntected component $C(a)$ you must be given a topology $\tau$
What's the topology on X in your exercise?
A: A set/space $A$ is disconnected when we can write it as $A=U \cup V$ where $U,V$ are non-empty, (relatively) open and disjoint. Being connected is then just defined as not disconnected. This is a correction to your assertions in the question.
Looking at all subsets of $X$ that contain $a$ is certainly an option (as the set is finite), but let's first observe that $X$ itself is not connected as witnessed by $U=\{a,b\}, V=\{c,d\}$. So connected subspaces must be smaller than $X$.  In your list of subspace topologies we see that $\{a,b,d\}$ has a disconnection $U=\{a,b\}, V=\{d\}\}$, so not connected either and $\{a,b,c\}$ likewise is not.
$\{a,b\}$ has the indiscrete topology as its subspace topology so is connected. Its clearly maximal by the previous considerations so the component of $a$ (and $b$ too) is $\{a,b\}$.
