Epistemology is a major branch of philosophy dealing with knowledge. While it can often get quite abstract, it will be especially important to my professional life - given I'm majoring in a scientific field.

• Cartesian Skepticism: Descartes was foundational in the field of philosophical skepticism. Essentially, he argued that we cannot really trust anything about reality; our senses are faulty, and even if they weren't, we could be being deceived by some all-powerful entity. Descartes thought of this deception as coming from a demon, but more modern takes might include the "brain in a jar" hypothesis or the simulation hypothesis. The only certainty, to Descartes, was of his own existence, as he can think, and therefore there must be a thinker - himself. It's hard to argue with this, but the problem comes with Descartes tries to "build back up" a foundation for reality, in which he appeals to God's existence (which he claims he can prove through only his thoughts). Most modern philosophers agree he failed in this respect, and I'm inclined to agree. So, what do we take from Descartes? Well, we can never be 100% sure of anything founded on experience with the outside world, not even of its existence, but it's simply infeasible to go around assuming reality doesn't exist, and as we learned from Descartes's failure, it's awfully hard to get back to sanity after doing so.
• David Hume: Hume is also considered a champion of skepticism, but his approach was much different, and in my opinion, more successful. Hume was an empiricist; he argued that knowledge and reason are not inherited from some divine spark, but merely evolved with humanity as one of our insticts. That doesn't mean it's bad - in fact, Hume was a major proponent of scientific thought. But at the end of the day, he views our faculties of reason as a useful tool, not something that is guaranteed to be right. When we measure the force of gravitation and find consistent laws, we do not know that those laws exist for sure - we merely have a lot of evidence on our side. It would be foolish to suppose that things regularly break those laws, since we have no evidence for that claim and quite a lot to the contrary, but it could still theoretically happen. (Sartre also expands on this point.) That's not to say there is nothing that can be known a priori, as per Hume's Fork - some things, like mathematical facts, are guaranteed to be true given a set of axioms - but these axioms must then be justified by experience, which gets us back to a posteriori knowledge.
• Immanuel Kant: Kant divides knowledge three ways - analytic a priori (statements derived from definitions), synthetic a priori (statements derived neither from definitions nor experience), and synthetic a posterior (statements derived from experience). This is largely in line with Hume's views, but I find the analytic/synthetic division of a priori knowledge to be a distinction without a difference. All a priori knowledge is, ultimately, derived from definitions; if you deep enough into a priori fields like mathematics, you find that you're left with definitions (axioms) and necessary consquences of those definitions, such that there could be no other conclusions.
• Thomas Kuhn: Of particular relevance to my field is the writings of Kuhn, a philosopher of science. He viewed knowledge aquisition (in other words, the scientific method) as a cycle of normal science (incremental improvements in measurements), intellectual crisis (contradictions arise in our knowledge), and paradigm shift (a new theory rises to resolve the crisis), and back again. This seems to be a relatively accurate description of science, and his view that science does not find the absolute truth, but rather seeks the theory that best fits our experience - meshes well with Hume's thoughts.

I think Hume's views on knowledge are the most practical, especially when combined with Kuhn's cycle of science. That which describes our experience the best is that which is closest to being true; in other words, effective theories are "more correct," even if there is no way to confirm them absolutely. Kuhn's writings also raises a more personal question - what approach do you take in life? An approach similar to normal science is choosing a lifestyle, and trying to refine that lifestyle into the best it can be. An approach more characteristic of paradigm shifts is trying many different possibilities, never sticking with one for long. This is similar to a problem in AI research known as the exploitation-exploration tradeoff, where exploiting known possibilities comes at the cost of exploring new ones. So far, my approach to life has probably been more similar to normal science, and I don't foresee that changing - but like in science, sometimes change is necessary. We'll see!