Potential PhD supervisor expressed interest, then didn't follow up about meeting. Should I continue with admissions process or move on? After a Skype conversation and several correspondences with an Irish professor, the professor wrote that he will take me as PhD student. 

Although we had agreed on which project to work on and we planned to discuss about research questions, we have yet to discuss about research questions and proposal. As I feel that he was really busy, I wrote him research questions to ask for his comment without having any prior discussion and mentioned that I need to make a decision if I am going to his University end of this month. Following that, he mentioned that I can put in Online application, will direct HR to advise me on Visa application, and maybe we would have another Skype conversation last Friday. In the email, he also sent me two papers and mentioned one research area of particular interest to be included as research questions.

However, the research area is really very new to me and so I told him in the last correspondence and ask him about time to have Skype conversation last Friday. I didn't hear from him since. 

I really interested to go to his university. Now, Should I put in online application first or I should forget about this opportunity and move on?
Don't panic.  The fact that you haven't heard back from him shouldn't be interpreted as a sign that he has lost interest in working with you.  It's summer, and academics often take vacations or travel to conferences in the summer.  It often happens that they take much longer to reply to emails.

Go ahead with anything that you already know you have to do, such as submitting the online application.  It doesn't sound like any urgent action is needed from the professor - you just want to chat with him about research when he has time - so just wait for him to respond.  If you don't hear anything within another week, send another email.