Could you please tell me about Mass Intentions?

The Holy Eucharist is the source and summit of the interior life and the very life of the Catholic Church.  Every time Mass is celebrated, the sacrifice of Our Lord and Savior Jesus Christ on Calvary is re-presented in an unbloody manner affording the priest celebrant and all congregants who participate in the Mass an infinite amount of sanctifying grace.  That grace can be applied to specific intentions, whether of the priest celebrant or the faithful.
Whenever a priest celebrates Mass, he has at least two intentions.  The first intention is to celebrate the Mass according to what the Church does.  The second is to apply the grace of that Mass towards a specific need, whether it be a person(s), living or dead, baptized or not, or a special need.  Every parish pastor is required, by Canon Law, to offer at least one Mass on Sunday for the needs and intentions of his parishioners (Can. 534 §1).
As to the first intention, a priest may find it helpful to  pray this traditional prayer while he is vesting: “My purpose is to celebrate Mass and to make present the body and blood of our Lord Jesus Christ according to the rite of the holy Roman Church to the praise of our all-powerful God and all his assembly in the glory of heaven, for my good and the good of all his pilgrim Church on earth, and for all who have asked me to pray for them in general and in particular, and for the good of the holy Roman Church.  May the almighty and merciful Lord grant us joy and peace, amendment of life, room for true repentance, the grace and comfort of the Holy Spirit, and perseverance in good works.  Amen.”
It is helpful for the priest to pray that prayer before Mass just in case he gets distracted at Mass and looses track of what he is doing.  At least he stated his intention before the Mass, and God takes that into account.
As to the second intention, commonly called “the intention of the Mass” and most likely what you want to know about, that intention is usually announced in the Church bulletin at least one week before, and then it is usually announced out loud at Mass.  Often the Mass intention is requested by a parishioner for the repose of the soul of a loved one.  When a parishioner requests that a Mass be offered for a specific intention, and the priest agrees to celebrate the Mass for that intention, then the priest has a most serious obligation to offer the Mass for that intention.  It is a good and pious custom for the faithful to make an offering, usually about 10 dollars, to the priest who celebrates the Mass.  The 10 dollars does not buy a Mass!  Masses and graces are not for sale.  Rather, the 10 dollars is a gesture of solicitude to the priest for his upkeep. 
In case you are wondering, a priest can not get rich on Mass offerings since he is limited to one Mass on weekdays, and two on Sundays, although in special circumstances he could celebrate two Masses on weekdays (that’s called bination) and three on Sundays.  The most he could make on Masses in a week is $150, or about $7,500 a year. 

It happens that the parish might receive more requested intentions than Masses are available.  In that case, the pastor can ‘bundle’ a bunch of intentions once or twice a week and load those intentions on to one Mass so long as the person requesting the Mass knows the Mass is for more than one intention.  In that case, no one loses out because there is an infinite amount of grace available at each and every Mass. (cf. Decree Mos iugiter obtinuit from the Congregation for the Clergy, 1991)