This is how I understand it:
Rick Webber was Leslie's husband and Laura's step-father. Laura took his last name at some point (not sure if Rick adopted her). Prior to marrying Rick, Leslie had an affair with David Gray and gave birth to Laura. She gave Laura to the Vinings to raise, so she grew up as a sister to Amy Vining. Laura and Leslie re-connected when Laura was a teenager. David Gray married a women named Florence, who we now know is Cyrus and Martin's mother.
Jeff Webber, Rick's brother, is Elizabeth Webber's father. Technically she and Laura are step-cousins. Had they been actual first cousins, Elizabeth and Lucky (Laura's son) would be too closely related to get married. However, they aren't related by blood so it wasn't an issue.