The 776308-1: It uses a 6.00 mm pitch and a 6.30 mm row spacing, which sets the board footprint and the mating half you need. You'll find it in automotive engine compartments, industrial control panels, and any harness that needs a positive-lock connection in a tight space.
Locking Ramp and Keying
When the housing is mated to its counterpart, the ramp snaps over a catch on the header — you should hear it click. That audible and tactile feedback tells you the connection is fully seated and won't work loose. The ramp also keys the housing so it can only go in one way; if it backs out, it's scrap.
Terminals and Mating Half
The mating half is a header or receptacle with female contacts that matches the 6.00 mm pitch and 6.30 mm row spacing. Confirm the terminal part number from the TE application drawing before you commit the BOM line.