Indicators will pop up adjacent to your general health status during certain conditions. This is a bad thing, so you should pay attention to them.
This symbol can only appear for people with a health implant. If you don't have one, you may want to keep a medical analyzer on hand to scan yourself when you see your health dropping for unknown reasons. If you seem to be gaining radiation rapidly, get to medbay as soon as possible. Radiation causes toxin and burn damage, radiation sickness, and may result in good or bad (mostly bad) mutations.
Blood is pouring out of your open wounds. As humans don't have an infinite supply of blood, you should ideally head over to medbay and ask for a bandage or suture, or close the wound in some other way. Otherwise, you will eventually pass out and die after losing a large amount of blood.
You are suffocating, generally because you're in space or hull breaches have rendered a part of the station uninhabitable. Put on some internals or move to a safer area. If you have a gas tank equipped and this pops up, make sure it's not empty or that you haven't set the pressure too low. By default, the lowest possible setting for 100% oxygen (emergency oxygen tank, jetpack) is 17 kPa. Air mix tanks require 85 kPa.
The biohazard symbol indicates that you're breathing plasma or possibly really bad farts. If it is the former, you'll take toxin damage as long as you do, so get some internals and leave the area before some idiot sets the air on fire.
This indicates that the air you are breathing is too hot and is literally burning your lungs. You will take burn damage very quickly as long as you expose yourself to this atmosphere. Generally a sign of a engineer pushing the engine to its limits and beyond. Quickly get your internals on and get out of there.