Sorry I'm coming to the party late; I hope you got your problem solved without incident.
This sort of thing can indeed be very dangerous. Although it's possible that the problem is with a connection to the power company's transformer (outside your house, which wouldn't put your house in danger), it's also quite possible it's a problem with a connection on the side of your house (i.e. the electric meter), or even a problem with a connection between the meter and your main service panel (fuse/breaker box). Either of those latter connections could be a serious safety issue; see this webpage for an example. (Note that though that website would like to blame the problem on a "smart meter", the problem in that case has nothing to do with the smart meter itself, instead it's a problem caused by the poor training of the workers who installed the smart meter.)
The reason half of a home's power can go out without affecting the other half is that residential homes in the U.S., Canada, and many other countries are served by two "legs" of 120 VAC whose waveforms are 180 degrees apart; combined, they provide an AC waveform whose voltage is 240 VAC. If one of the legs becomes disconnected prior to reaching the service panel, you can get all sorts of interesting results. Ideally, one half of the circuits in the house (the ones fed by that leg) will simply get no voltage. But there are circumstances that can lead to those circuits getting some reduced voltage: If a 240 V appliance is turned on, even though there isn't 240 V available to power it, voltage may (depending on the appliance's design) leak through it from the good leg to to the bad leg, providing some reduced voltage to the otherwise unpowered leg. In such a case you might see your incandescent bulbs glowing dimly; and if the 240 V appliance is a heater (oven, baseboard heater, etc), you may even get some small amount of heat from it because current will be leaking through it to any devices on the bad leg that are turned on.
As far as the danger of fire goes, as long as the power stays off you're probably ok. The real danger is that this is probably caused by a loose connection, and loose connections can get very hot when current is flowing through them. To be completely safe you can simply turn off your main breaker & call an electrician immediately. If you're even the slightest bit DIY savvy, you can look at your service panel and identify the circuits on the bad leg and turn them all off (and turn off all of the 240 V circuits). Service panels are almost always arranged such that adjacent "spaces" (one on top of the other) are on opposite legs, so that a 240 V breaker spans two "spaces" in the panel in order to connect to both legs of the service. Figure out some of the 120 V circuits that are on the bad leg, and turn their breakers off. You might begin to notice a pattern; the breakers that are off are all in either odd or even "spaces" COUNTING FROM THE TOP DOWN ONE SIDE. [Unfortunately most panels' spaces are numbered with odd numbers on the left and even numbers on the right; ignore the space numbers printed on the panel for the purposes of this discussion.] Once you've identified whether the bad leg is the odd spaces or even spaces, turn all of them off for that leg (odd or even spaces counting from the top down one side). [BTW: Sometimes you will find high-density small twin/tandem breakers where two breakers in a single package fit into a single "space" in the service panel. Note that those two breakers are both on the same leg, since one "space" provides power from only one of the 120 V legs.]
If you either shut off the main breaker, or shut off all the breakers on the bad leg, your risk of fire due to this problem is probably eliminated, if a fire hasn't already started. In any case, yes, you should call either an electrician or the power company ASAP to figure out where the problem is and fix it so you can turn your breaker(s) back on! (Just a suggestion: Because problems like this can be intermittent, which can make them difficult to troubleshoot, you might consider going outside to check the temperature of your meter and/or meter box before it has a chance to cool down, which can be helpful information for your electrician or power company service person... but be careful, if it's hot it could be very hot!)