Some examples to expand Andreas' answer might be of interest: (it's too much to fit in a comment so I'm adding it as an answer, though I think Andreas' response is great)

 - any metric space will be Frechet-Urysohn (choose $x_n$ in $A$ within $1/n$ of $p$); (more generally, any first-countable space is F-U: just choose $x_n$ in the intersection of $A$ and $U_n$ where $\{U_n\}_n$ is a countable base at the limit point);

 - a sequential but not Frechet-Urysohn space is given by taking $((\omega+1)\times\omega)\cup\{*\}$ where each copy of $\omega+1$ has the usual topology and a base for $*$ consists of sets
$A_{m,n}=\{(m,n)|m>M,n>N_m\}$
for $M,N_m\in\omega$ (ie cofinitely many elements of cofinitely many fibers) - then $*$ is in the closure of $\omega\times\omega$ but is not the limit of any sequence of points in $\omega\times\omega$; however, it is the limit of the sequence $x_n=(\omega,n)$ and each $x_n$ is the limit of a sequence of points from $\omega\times\omega$;

 - a countably-tight but not sequential space could be given by taking $(\omega\times\omega)\cup\{*\}$ where all points $(m,n)$ are open and a base for $*$ consist of sets
$A_{M,N}=\{(m,n)|m>M, n>N_m\}$
for $M,N_m\in\omega$ - this space is trivially countably tight (it's countable) but is not sequential: $*$ is not the limit of any sequence in $\omega\times\omega$ (since we can always exclude a cofinal subsequence of any putative sequence converging to $*$);

 - finally a non-countably-tight space is given by $\omega_1+1$ with the usual topology: $\omega_1$ (as a point) is in the closure of $\omega_1$ (as a set) but any countable subset of $\omega_1$ has bounded (countable) closure.