Board
Board is synonymous with plank: a single, usually rectilinear, length of wood sawn or riven (split) from a tree.
Generally the grain would run along the longer dimension of the piece, but a short but wide piece can still be called a board. Note that in certain woodworking traditions there are specialised additional words for boards of certain sizes or proportions, including narrow lengths or shorts pieces (although both come under the general term 'offcut' since that's literally what they are).
Panel
In furniture work panel might be used to describe any piece beyond a certain size, regardless of what it is made from and regardless of shape — basically any big, flat piece could be called a panel. So for example tabletops are panels, and one might say "I glued up three boards to make up the panel for the tabletop" and such a tabletop could be rectangular, oval, circular or another shape.
These days when working in solid wood wider panels are glued together from two or more boards for a number of reasons. In the past, where huge old trees were much more common than today, even very large panels could often be single pieces of wood simply because boards of that size were available and didn't have a particular price premium as wide boards do today.
There is also a type of construction called 'frame-and-panel', and within this context the central piece held by the frame is called a panel regardless of size. See the various sizes of the panels in the top, front and side of this tool chest:

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And can sometimes be very small, as in the doors of this spectacular one:

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I would appreciate if someone could tell me should a bent panel (90 degree bend along the center) is still called a panel since it's not flat any more?
I don't think a native speaker would describe something like this as "a panel", but there aren't any hard rules and many people talk informally and use their preferred (non-standard) words for things in English (both American and British English, as well as elsewhere).