The Church has identified the 4 main ways we can express our prayer as:
A - Adoration
C - Contrition (sorrow for sins)
T - Thanksgiving
S - Supplication (petition)
And the Church has also identified the 3 different types of prayer as:
vocal, eg. set vocal prayer such as Our Father
meditative, eg. Lectio Divina, meditation on Scripture
contemplative - the “loving gaze” between us and God
As you can see though, these are not clear-cut definitions, as in, our prayers when said naturally will probably incorporate many of these aspects. The Rosary for example is a vocal prayer, but at the same time invites us to meditate on episodes in Scripture, so both vocal and meditative. While all these types and methods have been identified, our prayer life is basically a relationship with God, so it will be like a rich tapestry of all these aspects woven together.