Is it acceptable for me to email the writer of the paper, and ask for explanations of basic concepts in the paper?
Not really, no. When somebody writes a paper about something, they're not offering to explain the basics of that field to every stranger in the world who might be interested. Rather, they're explaining their particular advances to people who are already expected to be familiar with the basics.
You should try to find other sources for the basics. Textbooks, websites, earlier papers in the same field, members of your own department, a relevant Stack Exchange site, and so on. If you can't find appropriate resources on your own, it would definitely be appropriate to ask the authors of the paper if they have any recommendations of resources you could use to learn about the area.
I'm assuming here that, by "basic concepts", you mean the basic concepts of the field, rather than the concepts that they introduce. For example, if the paper is about prime numbers, it wouldn't be appropriate to ask the authors what a prime number is, or what Euler's totient function is. If, on the other hand, their whole paper is about their new concept of "squiggle primes" and you can't follow their definition, it's reasonable to ask about that, if no more local person can help you. But do try to ask something specific rather than, "You define squiggle primes but I don't get it. Please explain more."