for free sms. Other than being practically the first search result I have no knowledge/experience/whatever with this service.
There are several messengers out there. Skype allows for free messages (including video) to other Skype users. And that goes worldwide, I speak out of experience. Skype is not alone, Practically all messengers allow you to send messages for free to other users of their software. And all will notify the receiver in the tray (on a computer) or message hub of some kind on a smart phone (Android/iOS/WindowsPhone). If there is a client available for the operating system your phone is using, you can communicate.
Google HangOut, Facebook Messenger are a couple of alternatives for Skype. There are many more. Communication between different clients won't work. For example: An attempt to send a Skype message to a person that only has Facebook Messenger, that will always be in vain. This applies to all of these messaging services. For Windows I know of 2 pieces of software 'Pidgin' and 'Trillian' that are able to pull partial intercommunication off, but most of the companies behind the messaging services are actively trying to block this.
The XMPP messaging protocol (also called Jabber) is the most open messaging protocol I know of. And there is actually some nice (and free) server software available. I personally like (and use) OpenFire
. If your internet provider allows for it (see the restrictions for the service you buy from them), you could set this server up and communication can commence through browser alone.
It would be handy, if you and the person you try to communicate with, have static IP addresses. Which is hardly ever the case with standard residential internet connections, unless you have a domain registered. There are ways to get a semi-permanent IP address/sub-domain (with limitations) for free, but this post is already getting long.
It cannot get less generic without detailed step-by-step explanation on how you want to communicate, what type of notifications should be used and on which devices all of this needs to be available on. Until then, your easiest and safest bet would be to have both you and the person you want to communicate with, use the exact same same messaging service on whatever devices you need this on. Then look each other up through this messaging software and create the connection between yourselves. Of course, both of you need to keep this software running 24/7, if you need/want that type of communication.