For an understanding of this post, please see my answer to How do the Pope, Bishops, Priests relate to one another in the Catholic Church's hierarchy?
Disciplining Delinquent Clergy
I will consider the practice of the Western Church, governed by the Code of Canon Law, since it is the one I know best. Since the Eastern Churches are governed instead by the Code of Canons of the Eastern Churches, there may be some slight differences (for example, the patriarch or major archbishop would have some oversight over his dioceses), but the procedures are essentially the same.
It is the responsibility of the bishop or major superior to discipline priests who commit crimes (whether civil or ecclesiastical). The proper procedures are outlined in Book VI of the Code of Canon Law. (There is a similar situation in United States: it is generally the responsibility of state governments, not the Federal Government, to prosecute criminals.)
However, in the case of very serious crimes, the so-called graviora delicta, the matter must be referred to the Holy See. It should, however, be kept in mind that not every civil crime is also an ecclesiastical crime, and the Church considers delicts to be “graviora” when they are likely to cause serious harm to the faithful.
Examples of graviora delicta include desecration of the Eucharist, violating the seal of Confession, and attempting to celebrate a Sacrament that is reserved to a higher rank of Holy Orders (for example, if a deacon attempts to celebrate Mass or hear a confession; or if a priest attempts an ordination; or even if a bishop ordains a bishop without a papal mandate). In the wake of the scandal regarding child abuse by priests, in 2007, the crime of pedophilia and child pornography were also added to this list. (For an excellent overview, see “Considerations on the Delicta Graviora” by Cardinal William Levada, then prefect of the Congregation for the Doctrine of the Faith.)
It is the responsibility of the Congregation for the Doctrine of the Faith (CDF) to process these more serious cases on the pope’s behalf. Of course, the CDF can only act when it is informed of a case, and so effective disciplining requires action on the part of the local church as well. Naturally, a bishop or major superior may (and should) act to remove a particularly dangerous priest or deacon from his assignment while his case is pending with the CDF.
In practice, at least in the United States, nowadays priests are removed from ministry as soon as credible allegations of child abuse of any kind are made, and their case is referred to civil authorities. They are not technically “suspended” unless they are found guilty. If they are found innocent (both by the civil authorities and by the internal Church investigation), they are generally reinstated to ministry; if guilty, they may be suspended—that is, permanently prohibited from exercising ministry—or laicized, depending on the seriousness of the crime. (Only the Holy See can laicize a priest, however, and so in that case the diocese would have to request it.)