So physical damage is the type of damage you do with auto attack and some abilities. The ability description will always tell you the type of damage it does.
Physical damage is countered by armour.
Some abilties do magic damage. Magic damage is pretty much the same as physical damage and it is primarily used by mages and other heavy spellcasters.
Magic damage is reduced by magic resist.
True damage is the least common one, and unlike the other two types it ignore resistances and damage reduction.
A champion like Talon is always going to do mostly physical damage, while a champion like Syndra is going to do mostly magic damage. The damage type is mostly important for tanks so that they can build to fight the enemy team, stacking a lot of magic resist vs magic damage team or armour vs physical damage teams.
The lightning you see is a keystone mastery called Thunderlords Decree. Every 3rd hit on a champion (With a 30-40 ish second Cooldown) does a blast of extra damage in a small area.