Kneeling at Mass is one of many postures during the liturgy. At different times we are to kneel, sit, or stand depending upon what is taking place during Mass. Each posture takes on a certain significance within the liturgy, especially kneeling.
Kneeling is a very humbling posture. The very act of kneeling before someone is to put yourself in a vulnerable position recognizing the other person’s authority, or when we kneel before God, it is that we are acknowledging his holiness and greatness. In the Catholic Church we believe that the bread and wine are transformed into the true body and blood of Jesus Christ although remaining under the appearance, taste, and texture of bread and wine. The process is called transubstantiation and what happens is a miracle of faith given to us by Jesus and passed down through his disciples.
After we receive the body of Christ and his precious blood we return to our pew and kneel for a period of time. We kneel for one primary reason: it is a sign of respect for the real presence of Jesus Christ. We are to kneel until the Eucharist is stored safely in the tabernacle.
The General Instruction of the Roman Missal (GIRM), which is the main document that prescribes how a Mass is to be conducted, makes very clear that we are to kneel after receiving the body and blood of Jesus Christ in the eucharist. When I go to Mass I occasionally see some people kneel for a short time and then sit down on the pew. While the GIRM does make exceptions for people with certain physical conditions, those who are physically able to kneel should. We don’t kneel because of the priest and we don’t kneel until we are done praying even though the distribution of communion continues, we kneel when the blessed sacrament is out of the tabernacle, is being distributed to the faithful, and we may return to a sitting position once the blessed sacrament is no longer out, when the tabernacle door has been closed, signaling the end of the communion rite. Kneeling until the priest sits down is not necessary, but is permissible.
After communion there is often a silence (sometimes very short) and the GIRM says that “the faithful [...] may sit or kneel during the period of sacred silence after Communion.” (GIRM 43). Otherwise, kneeling has taken place from the Agnus Dei (Lamb of God) up until this point. The diocesan bishop has the authority to decide if something other than kneeling will take place during the Mass. In most dioceses, kneeling during communion is the norm.