Beginning at 1st level, while wielding a rod, staff, or wand, you gain an implement bonus to all to-hit rolls you make when casting a spell or activating a spell-like ability. This bonus applies to melee touch attacks, ranged touch attacks, standard melee attacks, or standard ranged attacks. Divine casters may also use their holy symbol as an implement.
The implement bonus is equal to +1 at 1st level, and increases by an additional +1 per 4 class levels, dropping fractions, to a maximum of +9 at level 33 (i.e. +2 at 5th, +3 at 9th, +4 at 13th, +5 at 17th, +6 at 21st, +7 at 25th, +8 at 29th, and +9 at 33rd). Note that, if you multi-class into another class that also gets this bonus, the class levels stack for purposes of calculating this bonus.
If you are not actively wielding a rod, staff, or wand, you do not get this bonus.
Furthermore, casters can flick, twirl, or swoosh a wielded rod, staff, or wand to complete the somatic gestures of any spells they cast, in place of using their empty hand(s). That is, an implement counts as an empty hand for purposes of completing somatic gestures.
When wielding a magic staff in both hands, a caster need only have material components in a readily available storage location (like a Component Pouch). The staff will draw out and consume the components as part of casting the spell, eliminating the need for the caster to pause in wielding the staff to draw the component themselves. This only works if the component is in an accessible place that could normally be pulled from as part of casting a spell. If the component is tucked away in a (non-magical) backpack, stored in a portable hole, or left in the caster's bedside table back in town, the staff has no power to pull components from there.