Against the odds: This means to succeed despite facing many difficulties or unfavorable circumstances.
Against the grain: This means to go against what is usual or expected, or to act in a way that is contrary to one's natural inclinations.
Against the clock: This means to work or do something very quickly, often under pressure to meet a deadline.
Against the law: This means to do something that is illegal.
Against the backdrop of: This means to happen or be considered in relation to a larger situation or context.
Against all reason: This means to do something that is illogical or makes no sense.
Against the tide: This means to go against the prevailing opinion or trend.
To understand the meaning of "against the" in a specific context, you need to look at the rest of the phrase.