What is the difference between these two sentences?
I started this blog for communicating with other country's people.
I started this blog to communicate with other country's people.
With some corrections...
I started this blog for communicating with another country's people.
Here for communicating describes the purpose of the blog.
I started this blog to communicate with another country's people.
Here to communicate describes the reason you started the blog.
I would say the second example is more natural, though both would be understood OK. Either may be better, depending on context.