My brother and I play this game using 8 white checkers and 8 black checkers on opposite sides of the board. We take turns flicking the checkers across the board to knock the opponent's checkers off the board. Once all the opponents checkers are gone, you win that round. We called this game "Chinese Checkers", but apparently Chinese Checkers is a different game using a star shaped board rather than a square one. So, what is this game supposed to be called?
I just googled "flicking Draughts game" and the first result was a wikipedia page on a game called Chapayev
Details from the article include:-
The game requires a checkerboard and checkers, eight pieces of each colour.
Which matches your question.
During the first round, white pieces are placed on the first row, and black on the last.
Which fit requirement of "checkers on opposite sides of the board"
A player takes their turn by flicking one of his pieces with the index finger to shoot it at one or more of the opponent's pieces.
Which again matches your description
The game continues until only one colour remains on the board, winning the round.
Which matched the victory condition you describe.
With either two players or four players split into two teams, your goal is to shoot (or really, flick) discs into the center hole or higher scoring regions of the board, while also attempting to knock out the opposing team’s discs.
Without the center hole and pegs the game will naturally degenerate into simply attempting to flick the opponent's discs out of play.
Although not well known outside of Canada, Crokinole is a perennial favourite of kids and adults alike in Canadian summer cottages and games rooms.