It sounds like what you want to do is write an expression for your label instead of simply choosing a "Label" field.
On the Label tab of the Layer Properties:
Choose the "Expression" button:
Here, you can see that I have entered the following: "Parcel Number = " & [APN]
The first part of this: "Parcel Number = ", is simply a text string
The second part: [APN], refers to an attribute field, and will pull the value from it.
If you click on the "Verify" button, it will show you how this expression evaluates.
This can be used to combine different text strings and attribute fields, as well as write conditional expressions. Please see the help section on Essential Labeling Concepts.
There are many ways of configuring labels in ArcMap, and the help sections do a good job of describing them and providing examples.