"Continuous" refers to the values your random variable can take: a discrete set like the integers, or a continuum like the real numbers? "Cumulative" means that the density function measures $P(X\leq x)$, the probability that $X$ is in the range of values up to and including $x$ (like how a cumulative exam covers everything you've learned up till now).
To recover the probability density function (pdf) from the cumulative density function, you either need to differentiate (if the random variable has a continuous distribution) or take a difference (if the random variable is discrete).
In particular, in your example, the distribution looks discrete since values for the cumulative density function are only given for integer values of $x$. (You can tell it's "cumulative" because the values increase to $1$ instead of summing/integrating to $1$.) So to find the probability density function $P(X=x)$, you can find it by taking the difference in two neighboring values of $F$: $\begin{align*} P(X=x) &= P(x-1 < X \leq x)\\ &= P(X\leq x) - P(X\leq x-1)\\ &= F(x) - F(x-1)\\ &= \left(1-\frac{1}{2^{x-1}}\right)-\left(1-\frac{1}{2^{x-2}}\right)\\ &= \frac{1}{2^{x-2}} - \frac{1}{2^{x-1}}\\ &= \frac{1}{2^{x-1}} \end{align*}$ In general, to find $P(a< X\leq b)$ from the cumulative density function, just note that $X\leq a$ and $a< X\leq b$ are disjoint events whose union is $X\leq b$, so $P(X\leq a) + P(a < X\leq b) = P(X\leq b)$ $F(a) + P(a $P(a< X\leq b) = F(b) - F(a).$
I'll leave it to you to find the value of $P(10 in this example.