Internet connects disconnects when I switch on/off my fan from any of the room

Get a UPS for your router (and cable/dsl modem, if any).

Electric motors have big power draws. Turning them on or off can adversely affect the quality of the electrical power to other devices.

It's also possible that electrical outlet for your router is one that is tied to the light switch so that when you turn it off, you kill power to the outlet.

Is your broadband connection going through a modem and or router, or directly to the computer?

If it goes through a modem or router, the electrical problem is located where either of these items are plugged into electricity. If you are able to, try moving the plug for these to a different outlet location and see if it solves the problem.

I'm betting that it won't because as someone earlier said, the motor for the fan is drawing a lot of electrical current on that circuit. You may have to move the modem to an entirely different part of the house on a different electrical circuit. You should also have the electricity capacity in your home evaluated by a licenced electrician.

I've also seen a case of a light on a dimmer switch completely killing a DSL signal.

It was a floor lamp with a halogen bulb. Turn the light on full, and there was no problem. Turn the light off, and there was no problem. Use the dimmer at all, and suddenly no Internet.

Definitely related to the electrical circuit, because we tested by running an extension cord to the neighbor's house and leaving the lamp in place, and the DSL worked. Plugging the power cord back into the house current caused the same symptoms. Reversing the experiment (physically moving the lamp, but using an extension cord to the house) reproduced the symptoms as well.

We used to have a similar problem in one of the offices I did tech support for. By process of elimination we figured out that it was the copier that was drawing all the electricity anytime someone made copies. It caused numerous problems with the power in the server room. They put in a dedicated circuit for the copier and that solved the problems with both the computers and telephone system.

Rheostats/dimmer switches/fan speed controls can put a lot of RF or line noise out in the ether and back onto the electrical service. Likely scenario is that's what's clobbering your connection. I'd try moving the power cord of the router to a different circuit. If that doesn't work or is impractical, use a different fan if you need to keep cool.

While it's possible that the fan is somehow drawing too much electricity, it's not that likely to be the cause. It's more likely to be RF or line noise the fan's putting out and messing up the connection. These problems are not uncommon. Typically, I would use a line filter if one can be used on the power strip for the router and keep the hash from clobbering it.

This from the original post leads me to think that he is referring to something like central air or swamp cooler.

Well, there is some problem in whole circuit for sure but @DrewDad is right. Using UPS is the quickest solution for this problem.