This question relates more to the vim-airline plugin than to vim itself. I only mention that because it is easier to answer a question that is well-defined :P.
The vim-airline docs describe how to format buffer names.
Step by Step
Basically create a file named
that contains the following:
function! airline#extensions#tabline#formatters#foo#format(bufnr, buffers)
The function above would return the full file path I believe, but you can modify it to return any string you want. Once that is done, add this next line to your
let g:airline#extensions#tabline#formatter = 'foo'
And you should be good to go from there.