Mark each statement as TRUE, FALSE, or UNKNOWN
(a) $|\Bbb{R}| < \aleph_1$
(b) $|\Bbb{R}| = \aleph_1$
(c) $|P(\Bbb{R})| > \aleph_1$
Could someone explain to me the reasoning based on whatever the answer is for each one because I do not fully understand Countable Infinite and Uncountable Infinite sets
$\aleph_1$ by definition is the first uncountable cardinal number. So $|A| < \aleph_1$ by definition means that $A$ is a countable set. The reals are not countable.
We (should) know that $|\mathbb{R}| = |P(\mathbb{N})| = 2^{\aleph_0}> \aleph_0$. So in particular $2^{\aleph_0} \ge \aleph_1$. The so-called continuum hypothesis (which is independent of ZFC, so "unknown" in your list) states that $2^{\aleph_0} = \aleph_1$.
By Cantor's theorem $|P(\mathbb{R})| = 2^{|\mathbb{R}|}= 2^{2^{\aleph_0}} > 2^{\aleph_0} \ge \aleph_1$.