If you text someone who has an iPhone, then there are no records of that text with the cell phone carrier. Texts sent through iMessage from iPhone to iPhone show as a "blue bubbles", i.e. the background of the text is blue.
If on the other hand you text to someone who doesn't have an iPhone, the text is sent as SMS through your cell phone carrier which will have a record of it and the number you texted may appear on the carrier statement. Texts sent this way show as "green bubbles", i.e. a green background.