Upgrading equips simply require a core and a right click, it's that simple.
Low core = anything up to aramid level
Med core = anything up to titanium level
High core = anything up to red osmium level
Highest core = anything
Every time you succeed in upgrading, you gain +3 defense and +3 defense rate on your armor, +3 attack and +2 magic attack and +? on your attack rate on one hand sword type weapons, +6 attack and +2 magic attack on two hand sword type weapons, and +2 attack and +3 magic attack on magic weapons. The maximum + on equips is +7, and the higher you go, the shinier it gets. Also, +6 and +7 add double the stats previously listed.
+0 to +1 - Fail = nothing happens
+1 to +2 - Fail = nothing happens
+2 to +3 - Fail = nothing happens
+3 to +4 - Fail = nothing happens
+4 to +5 - Fail = -1 to +3
+5 to +6 - Fail = -1 to +4
+6 to +7 - Fail = -6 to +0
*Items can NEVER break.
So, be very very sure of your decision when you upgrade to that +7 rank, because it fails more often than not.
To upgrade an equip, simply right click the core, and then drag it onto the item you wish to upgrade, and click on it. Be aware that the stat requirements of your equip will increase when you upgrade it as well.