How do I open the blue gate in Chapter 8 (Core)?
That gate has 4 empty heart images, which means you have to collect 4 Crystal Hearts to open it.
There is a blue Crystal Heart hidden somewhere in each of the previous levels, and another red heart at the end of each B-side level, for a total of 14. If you collect any 4 of those, the gate will open.
See Also: Celeste Crystal Heart Locations and Solutions Guide