A professor at a university replied to my PhD application email with a yes. I asked him further about his current research projects but haven't heard from him for nearly 2 weeks now; should I write him a reminder email? if so, how can it be in the politest way possible?
I would recommend sending out that reminder email. As a prof, I get about about 20-30 emails per day (of course, many people get plenty more than that). After a week if I haven't responded to an email it is usually not because I am ignoring it. Rather, it has been drowned out by hundreds of more recent notes and messages. Reminders are welcomed -- in fact they tell me that a potential PhD student is both organized and values knowing about my research.
Perhaps a simple response like 'I am following up on my email from May xx, 2016. I assume you are very busy but have you had an opportunity to consider my request?' with your previous email included within the email as a reply.