How to calculate the number of possible connected simple graphs with $n$ labelled vertices

There are $\binom{n}2=\frac12n(n-1)$ pairs of distinct points. If you do not allow loops or multiple edges, each of these pairs determines one possible edge, and you can have any subset of those possible edges. A set with $\binom{n}2$ members has $2^{\binom{n}2}$ subsets, so there are $2^{\binom{n}2}$ possible graphs without loops or multiple edges.

If you demand that the graphs be connected, the problem becomes very much harder. From your final comment I take it that you are in effect counting labelled graphs. This sequence of numbers is A001187 in the On-Line Encyclopedia of Integer Sequences. If $d_n$ is the number of labelled, connected, simple graphs on $n$ vertices, the numbers $d_n$ satisfy the recurrence

$$\sum_k\binom{n}kkd_k2^{\binom{n-k}2}=n2^\binom{n}2\;,$$

from which it’s possible to calculate $d_n$ for small values of $n$. This recurrence is derived as formula (3.10.2) in Herbert S. Wilf, generatingfunctionology, 2nd edition, which is available for free download here.

According to MathWorld, Brendan McKay’s software package nauty includes a routine that efficiently enumerates such graphs; it’s available here.

If you count unlabelled graphs instead, so that you don’t count isomorphic graphs separately, you get the sequence mentioned by Arturo in the comments.

You can use exponential generating functions for this. The exponential generating functions $C(x)$ for connected labeled graphs and $D(x)$ for all labeled graphs are related by

$$D(x)=\mathrm e^{C(x)-1}\;,$$

which you can show using the decomposition of a labeled graph into its connected components.

As others have noted, we have

$$ D(x)=\sum_{n=0}^\infty\frac{2^{n(n-1)/2}}{n!}x^n\;, $$

so

$$ \begin{align} C(x) &= 1+\log\sum_{n=0}^\infty\frac{2^{n(n-1)/2}}{n!}x^n \\ &= 1+\log\left(1+x+\frac2{2!}x^2+\frac8{3!}x^3+\frac{64}{4!}x^4+\dotso\right) \\ &= 1+x+\frac1{2!}x^2+\frac4{3!}x^3+\frac{38}{4!}x^4+\dotso \;. \end{align} $$

Thus there are $1,1,1,4,38,\dotsc$ different connected graphs on $0,1,2,3,4,\dotsc$ labeled vertices.

As I indicated in the comment, this is sequence A001349 in the On-Line Encyclopedia of Integer Sequences. There is no closed formula listed, but there are a couple of references to computer calculations and algorithms. I suggest that's the place where you want to start.