A connecting vowel is an omicron or epsilon added to the stem of a Greek verb. See the master verb chart (BBG page 354) for what verbs require it and which don’t.
More technical information is:
- The mark of the subjunctive mood is a lengthened connecting vowel.
- If the first letter of the verb ending is a μ or ν, then the connecting vowel will be omicron. In all other cases, the connecting vowel will be epsilon.
- If there is no ending, then either can be used. See the chart on BBG page 352 for confirmation of this.