The Diet session is usually 150 days long.
Meetings are held every day during the session.
The Diet extended its session by one week.
The regular session begins in January.
The budget bill was deliberated during the session.
The Emperor convened the Diet session.
会議 means the meeting or conference itself, whereas 会期 means the scheduled time span over which that meeting or session takes place.
Compound of 会 (kai, 'meeting') and 期 (ki, 'period'). The exact historical derivation is uncertain, but the kanji straightforwardly indicate a meeting period.