Name Field is used to specify the name of the Group to search for. Owner Field lets you search a Group using it's Owner name.
Register dropdown says how other users are allowed to join the group:

• No One means no other user can join the group (you can still invite other users).
• By Request means that other users can request the group owner to join the group.
• Anyone means all users are allowed to join the group.

It should be noted that the root account can always join any group. The root account can also always take over ownership of any group.
The Access dropdown controls how users who belong/subscribe to a group other than the owner can access that group.

• No Read means that a non-owner member of the group cannot read or write the group news feed and cannot read the group wiki.
• Read means that a non-owner member of the group can read the group news feed and the groups wiki page.
• Read Comment means that a non-owner member of the group can read the group feed and wikis and can comment on any existing threads, but cannot start new ones.
• Read Write, means that a non-owner member of the group can start new threads and comment on existing ones in the group feed and can edit and create wiki pages for the group's wiki.

The access to a group can be changed by the owner after a group is created.

• No Read and Read are often suitable if a group's owner wants to perform some kind of moderation.
• Read and Read Comment groups are often suitable if someone wants to use a Yioop Group as a blog.
• Read Write makes sense for a more traditional bulletin board.