The introit in the OF is the Entrance Antiphon which usually should be recited when there is no music. When music it is not required to be said but technically it could be before the sign of the cross if the priest chooses too.
The Introit
is music when Gregorian chant is used, as still happens in many places in the OF, including the abbey where I attend Mass every Wednesday and Sunday. The appropriate Introit to use is in the Graduale Romanum, 1974 edition. Usually the vernacular un-sung introit is a translation of the one in the Graduale, which is in Latin.
To piggy back off of what porthos11 said, the Invitatory is the opening psalm of Matins and never changes (while its antiphon does). It’s always Psalm 94 and it’s split up into five parts with the antiphon interspersed.
Actually, psalms 23, 99 and 66 may also licitly be used as invitatory psalm, in the Liturgy of the Hours. I use 94 on Sundays, feasts and solemnities, and 66 on weekdays, because in the monastic tradition, Ps. 66 was the first psalm at Lauds every day, chanted in directum. It was always followed by psalm 50. So Ps. 66 would be particularly appropriate on Fridays if Lauds was the first office of the day, as juxtaposing psalms 66 and 50 is an ancient liturgical tradition that goes back to the Rule of St. Benedict, i.e. 1500+ years!
There are also post-Conciliar variants to the Monastic Office. The schema used at the abbey I’m attached to as oblate, uses a different invitatory psalm every day: 94 on Sundays, feasts and solemnities, 28 on Monday, 66 on Tuesday, 45 on Wednesday, 23 on Thursday, 8 on Friday and 80 on Saturday.
The Gradual also contains the Offertory and Communion antiphons.
It also contains the Gradual (which licitly can replace the responsorial psalm in the OF), the Alleluia with its verse, the tract for use in Lent in lieu of the alleluia, sequences where they still apply (e.g. Corpus Christi, Easter, Pentecost), the Ordinary including Credo, and various tones for responses, the Lord’s Prayer, etc. It’s what our abbey uses for the entirety to Mass with the exception of the Lord’s Prayer which is chanted in French. What is not in the Gradual, is chanted in French plainchant.
Also the Good Friday liturgy is in there, in Latin and Greek.