"She is the woman that I love."
Translation:Sie ist die Frau, die ich liebe.
"Dame" is a rather antiquated word and almost nobody uses it anymore if they don't want to make a point. There are still old words around like "Damenrad", "Damenbekleidung" and terms like "Meine Damen und Herren,..." etc. But nowadays, feminism has almost banned the word. It's not an insult, don't worry, but it does imply rather old-fashioned gender stereotypes whereas "Frau" is neutral. Hardcore feminists might actually be highly offended if you call them "Dame" but it can also be conceived as polite and gentlemanly, depending on the personal view of the person, I guess.
You should also be aware of the adjectives "dämlich" which means "dumb" and "herrlich" which means "wonderful". I know some women (and men!) who spit fire if you use either word...