A Base Competency (BC) determines the modifier applied to rolls in that speciality; calculated by adding the speciality in question to half its base attribute. I.e., with Melee +5 and Strength 6, a character would be said to have a Melee BC of 8: 5 from Melee, 3 from the Strength. This translates to a +80 to-hit bonus when making attacks. When halving an odd number, always round up!
BCs are sometimes denoted in the format of Melee|STR, especially for the purposes of calculating damage; i.e., such a notation calls for the addition of all of the character's Melee, and half of their STR. The "|" symbol is used to denote BCs, the "/" is used to mean that either stat could be used, and a "+", unsurprisingly, means straight addition:
A weapon with a +Melee|STR damage modifier would deal +8 damage in the hands of the character above, a weapon with a +STR modifier would deal +6 damage, and a weapon with a +Melee+STR modifier would deal +11 damage.
BCs also determine the effectiveness of various innate abilities and aspects; unarmed damage in Melee, for example, can vary from 1d2 to 1d20 depending on the user's BC, as can the damage of psionic characters.