I think this is bit round-about explanation but still it clarifies using real life example.
Small Example:
Let's say playing an audio involves three steps:
If your audio player does step 1,2,3 sequentially for every song then it is synchronous. You will have to wait for some time to hear the song till the song actually gets fetched and decompressed.
If your audio player does step 1,2,3 independent of each other, then it is asynchronous. ie. While playing audio 1 ( step 3), if it fetches audio 3 from harddisk in parallel (step 1) and it decompresses the audio 2 in parallel. (step 2 ) You will end up in hearing the song without waiting much for fetch and decompress.