There are only a few things that can block a chimney. Many chimneys are not built correctly to begin with, so they will send smoke back into the room even when they are not blocked. But here are a couple of tips to help determine which it is.
1. Did the chimney work previously? Is this a chimney that has always worked before, then one day started smoking? If so, you could have a blockage or;
2. Your damper may not be working properly. The damper is the door above the fireplace that you open and close to use the fireplace. Check it to see it is open properly. You will have to get down on your hands and knees with a flashlight, stick your head into the fireplace, and look up. Sometimes the damper door will come off it’s track or hinges (depending on the design), and it will need to be put back to function correctly.
3. While you are down on your hands and knees, see if you can look up past the damper to the flue pipe. Can you see all the way to the top of the chimney? If so, you know it isn’t blocked. But if you can’t see all the way up, it doesn’t necessarily mean it’s blocked. It may be because the chimney curves or is offset to the back of the damper so there is not a clear sight line.
4. If you are lighting a fire in a house you are new to, it is recommended that you get a qualified Chimney Sweep to inspect the chimney before you light it. But if you want to do it your self, then try this. Wad up 4 or 5 pieces of newspaper, put them into a pile in the middle of the firebox and light them. That way if there is a blockage and smoke comes into the room, the fire will go out quickly.
5. If smoke is coming out into the room, go outside real fast and see if there is any coming out of the chimney outside. That way you’ll know if it’s a total blockage.
What can block the chimney? The most common thing is a squirrel nest. Squirrels pack the chimney completely full of sticks and leaves to build their nests. It will totally block the chimney draft.

Believe it or not, I have come across chimneys almost totally blocked by spider webs! There is a type of spider that makes a web that is more like gauze than the usual geometric pattern. This type of web can totally shut off air flow in a chimney.
Either way, the chimney will need to be cleared by a professional.
There can be other causes of blockage, including bricks falling into the chimney on older chimney stacks. So it is in your best interest to have someone who knows what they are doing look at it. That would be Chimney Sweep Pros.