Your Smite would resolve first, The structure of turns and actions is roughly as follows:
Beginning Phase - [Untap step, Upkeep step, Draw step]
Main Phase I
Combat Phase - [Beginning of Combat step, Declare Attackers step, Declare Blockers step, Combat Damage step, End of Combat step] Main Phase II
Ending Phase - [End step, Cleanup step]
So using this your opponent announces that their 2/2 creature is attacking during the Declare Attackers step, then when you get to the Declare Blockers step you declare your 1/1 as a blocker. Generally at the end of each phase there is a round of priority where players get the chance to activate abilities or cast spells, so during the end of the Declare Blockers step you cast Smite, before you can move to the Combat Damage step Smite must finish resolving, so therefore by the time you creature would take damage from your opponents attacker, it would be destroyed as long as it didn't have indestructible or was not regenerated.
March 16, 2016 12:57 p.m.
Addendum: I meant to say "...generally at the end of each Phase and Step".
March 16, 2016 12:58 p.m.
Raging_Squiggle says... #4
Note that if the attacking creature was regenerated, it would be removed from combat and would not deal any damage anyways.
March 16, 2016 1:53 p.m.
VaraSatoshiOfficial says... #5
If you cast Smite before the damage step, Smite would resolve in between the declate blockers and the damage step, meaning that the 1/1 would survive.
Raging_Squiggle says... Accepted answer #1
At the end of the declare blockers step, you have the opportunity to cast Smite before proceeding to the combat damage step. When that resolves, the "blocked creature" will be destroyed.
When in the combat damage step, your 1/1 will survive because there will be no attacking creature to deal damage to it. Likewise your 1/1 will not deal any damage because the creature it blocked is no longer on the battlefield.
March 16, 2016 12:51 p.m.