The Catholic Church is one of several denominations that teach the Real Presence, but they don't all have the same understanding of this doctrine. What is the Catholic understanding?
For my part, I can quote the inside flap of the Missal in the pew which says, Catholics believe that the Bread and Wine become the Body and Blood of Christ (not a symbol, but actual fact).
It goes on to say, the Catholic Church permits eastern Orthodox adherents to come to Catholic Mass, celebrate the Eucharist and receive communion. But Catholics are not permitted to go to their Mass unless necessary.
Other Christians are excluded from reception of the Eucharist, mainly because if they hold strong to their professed belief, they do not believe the Eucharist could be what Catholics say it is. (They may believe that it is the Body and Blood of Christ, but they most certainly don't believe that only a validly ordained Catholic Priest could consecrate the host).
All peoples of all religions are invited to come to Mass and participate in the Liturgy.
Transubstantiation means that the Stuff of the bread and wine is totally replaced with the Body and Blood of Christ. Catholics don't believe that the wine is His Blood and the bread is His Body, Catholics do believe that what can be seen after consecration is just plain old different, even though there is no perceptible change. In this view, Catholic and Orthodox is the same.
In the Orthodox Church is it common to intinction (dip the consecrated bread in the consecrated wine) - but it doesn't essentially change the belief.