synchronous
When I call you on the phone, I dial your number and WAIT until you pick up. Then you say something, and in the very same moment I listen to you. When you finished, I send you data (talk to you) and in the same moment you receive them (listen to me). At the end of our communication one of us says "END OF TRANSMISSION" (Good Bye), the other says "Acknoledged" (Good Bye) and then both ring off.
asynchronous
I write you a letter. I put it to the postoffice, and it will be sent to you. In the meantime I do NOT WAIT. I do many different other things. Then you receive the letter. You read it while I still do many different other things. Then you write me an answer and send it to me. In all those things I am not involved. At the next day I get a (synchronous) message (a signal) from the system (postman). It (he) says: "Here is a message for you". Alternatively I could poll my inbox every five minutes to check if a new letter is there. Then I pause my other work, receive your letter and read your answer. Then I do something according to this answer. But this are things you will not notice, because you are not involved in what I do with your asynchronous answer.
In many circumstances they are different names for the same thing, but in some contexts they are quite different. So it depends. Terminology is not applied in a totally consistent way across the whole software industry.
For example in the classic sockets API, a non-blocking socket is one that simply returns immediately with a special "would block" error message, whereas a blocking socket would have blocked. You have to use a separate function such as select or poll to find out when is a good time to retry.
But asynchronous sockets (as supported by Windows sockets), or the asynchronous IO pattern used in .NET, are more convenient. You call a method to start an operation, and the framework calls you back when it's done. Even here, there are basic differences. Asynchronous Win32 sockets "marshal" their results onto a specific GUI thread by passing Window messages, whereas .NET asynchronous IO is free-threaded (you don't know what thread your callback will be called on).
So they don't always mean the same thing. To distil the socket example, we could say: