To resolve is to settle or make a decision about something — often formal. A college's board of directors might resolve to recruit more minority students. As a noun, resolve refers to a strong determination to do something.
If you make a New Year's resolution to exercise every day, you'll need plenty of resolve to stick with your program. The verb descends from Middle English resolven "to dissolve," from Latin resolvere "to untie." In English, the obsolete sense of "to dissolve" can be seen in this line from Shakespeare: "O, that this too too solid flesh would melt, thaw, and resolve itself into a dew."