The intersection of all the Sylow $p$-subgroups is called the $p$-core of $G$ and is the biggest normal $p$-subgroup of $G$. In other words, if you have a normal $p$-subgroup in $G$, then this subgroup is contained in all the Sylows. Easy example: dihedral group of order 20. The centre is of order 2 and is contained in every Sylow 2-subgroup.
This is not the only way that two Sylows can intersect. The intersection of two Sylows can be non-trivial even if the $p$-core is trivial. For example in $A_{10}$, the Sylow 5-subgroups are each generated by 2 disjoint 5-cycles. The groups $\langle (1,2,3,4,5), (6,7,8,9,10)\rangle$ and $\langle (1,2,3,4,5), (6,7,8,10,9)\rangle$ intersect in a cyclic subgroup of order 5. But the 5-core is trivial, since $A_{10}$ is simple.