Within the context of the Christian church, a dogma refers to a tenet that is considered authoritative and binding.
The word is of Greek origin, and in pre-Chrisitan context was used to mean governmental decrees (see, e.g., the decree of a census made in Caesar Augustus's time, Lk 2:1) or principles of philosophy or science. It is also in this sense that it is used in Acts 17:7; Eph 2:15 and Col 2:14. In the NT, it is used to refer to the decision made by the apostles at the Jerusalem Council that Paul and Silas delievered to the churches (Acts 16:4).
The word is used in various ways throughout the history of the Church. During the early centuries it was used rather loosely to refer to almost any and everything that pertained for faith. Eventually it became more definite to mean only those articles of faith that define what is authoritative and essential.
Dogmatics refers to the product derived from the collection, preservation and exposition of the tenets of the Christian faith, or what is often called 'systematic theology'.
The use of the word is, of course, not confined only to the Church. Generally, a dogma refers to any tenet that is held to be authoritative, whether this is in the field of science or philosophy.
Resources
Louis Berkhof, Introdcution to Systematic Theology. 
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