The Laplace transform converts a real valued function, $f$, from the real valued time domain, $t$, to its transform, $F$, in the complex domain, $s=\sigma+j\omega$.
$$ F(s)=\mathscr L\{f(t)\}=\int_0^\infin f(t)e^{-st}dt $$
The inverse Laplace transform is technically given by the following, though in practice, partial fraction decomposition and tables are generally used to perform both Laplace transforms and the inverse operation.
$$ f(t)=\mathscr L^{-1}\{F(s)\}=\frac 1{2\pi j}\int_{c-j\infin}^{c+j\infin}F(s)e^{st}ds $$
The Laplace transform has a few features which make it useful. First off, it is a linear transformation, such that $\mathscr L\{af_1+bf_2\}=aF_1+bF_2$. Secondly, Laplace transforms of integrals and derivatives turn out to simply be algebraic manipulations of the Laplace transform of the base function. As such, ODE problems can be converted to algebra problems. Thirdly, there is a consistent pattern for shifting in the time and complex domains. Fourthly and finally, the final value theorem gives steady state time domain behavior in terms of initial Laplace transform behavior.
$$ \lim_{t\rarr\infin}f(t)=\lim_{s\rarr 0}sF(s) $$
A comprehensive Laplace transform table may be found from Paul’s Online Math Notes