Task Analysis is:
"...the process of learning about ordinary users by observing them in action to understand in detail how they perfrom their tasks and acheive their intended goals. Task analysis helps identify the tasks your website and applications must support and can also help you refine or redefine your site's navigation or search by determing the appropriate content scope."
"Task Analysis". From http://www.usability.gov/how-to-and-tools/methods/task-analysis.html
A Use Case:
"...is a written description of how users will perform tasks on your website. It outlines, from a user's point of view, a system's behavior as it responds to a request."
"Use Cases". From http://www.usability.gov/how-to-and-tools/methods/use-cases.html
"...represents a discrete unit of interaction between a user (human or machine) and the system...each Use Case describes the functionality to be built in the proposed system..."
"The Use Case Model". From http://www.sparxsystems.com/resources/tutorial/use_case_model.html
A Contextual Interview:
Is an interview in which "...researchers watch and listen as users work in the user's own environment...and [the interviews] don't use tasks or scripts".
"Contextual Interview". From http://www.usability.gov/how-to-and-tools/methods/contextual-interview.html
A Contextual Inquiry:
"...is a semi-structured interview method to obtain information about the context of use, where users are first asked a set of standard questions and then observed and questioned while they work in their own environments."
"Contextual Inquiry". From http://usabilitybok.org/contextual-inquiry
A Focus Group:
"...is a pointed discussion with a group of participants that a moderator leads through a set of questions on a particular topic to obtain feedback about users, products, concepts, prototypes, tasks, strategies, and environments."
"Focus Group". From http://www.usability.gov/what-and-why/glossary/focus-group.html