What does it mean to say a random variable is non-negative?

$X$ is non-negative just means that $P(X<0)=0$. The opposite of "non-negative" is not "negative," just that the random variable might take a negative value, that is $P(X<0)>0$.

A "negative" random variable is one that is always negative - that is: $P(X<0)=1$. Similarly, for "positive," $P(X>0)=1$. Note that a positive random variable is necessarily non-negative. But a non-negative random variable can be zero.

A non-negative random variable is one which takes values greater than or equal to zero with probability one, i.e., $X$ is non-negative if $\mathbb{P}(X \geq 0) = 1$.

A negative random variable is one which takes values less than zero with probability one, i.e., $Y$ is negative if $P(Y < 0) = 1$. An example would a random variable which is equal to $-1$ with probability $1/2$ and equal to $-6$ with probability $1/2$, or if $Y \sim \operatorname{Exponential}(\lambda)$ then $-Y$ is a negative random variable (since $Y$ is a positive random variable).

Note in particular that saying a random variable is non-negative is not the opposite of saying it is negative.

Suppose your random variable is your net return in dollars on a game in a casino.

If you pay money to play and lose it all (or lose part of it) the variable would be negative.

If you win more than you bet, your return will be positive.

Conceivably, if the game is rigged for you to always lose, all of the possible (nonzero probability) outcomes could result in you losing money. That could be called a "negative random variable".