Aside from their sensitivity to heat rats are very hardy, and rarely suffer infections aside from the occasional cold or abscess. When they do a have a cold and get a runny nose and/or eyes the discharge is reddish - this looks like blood, but is really just a red pigment called porphyrin.
Respiratory infections can develop into something dangerous - pneumonia or similar. See section on respiratory disorders for suggested treatments.
They are, however, quite prone to skin problems. Scabbing which starts around the mouth and ears and then spreads to the back is usually due to ringworm or other fungal infection, which can be treated with a dipping solution, e.g. Imaverol (from vet). Imaverol should be made up as a solution of one part Imaverol to fifty of warm water and applied four times at three-day intervals: in extreme cases you may have to make up a bowlful and dip the entire rat, but it is usually sufficient just to dab it onto the affected areas with cotton-wool.
Scabbing which starts on the back and stays there is usually due to mites. Except in sterile lab. conditions all rats (and all humans, for that matter) have a few mites; but some rats, especially ones with pale coats, seem to be extra-sensitive to them, and some - especially those who are in poor health, or too infirm to groom themselves properly - develop an excessively high population. Mites can be treated with a dipping solution or with e.g. Ivermectin (from vet), a.k.a. Ivomectin or Ivomec. This is given in tiny amounts (and must never be given in large ones as it can damage the liver) by mouth or by injection, twice, two to three weeks apart. Some people also find that hanging an insect-repellant in the room gets rid of their rats' mites - but make sure it isn't something that will be poisonous to the rats themselves.
Some older rats just seem to get scabs because they cut themselves with their claws without noticing (most rats have somewhat insensitive skin - which makes giving them injections easier). Other possible causes of scabbing are dusty bedding or woodshavings; allergy to bedding or woodshavings; and too much protein in the diet.
They can also develop warts, usually on the tail: these look like small pink rosebuds, and usually fall off on their own after a few months.
Abscesses are also common. It is usually better to let these come to a head naturally and then clean out the crater and dress it.
Most abscesses are harmless and easy to deal with. However abscesses around the head - especially behind the eye - are a very bad sign. They usually mean that the animal's immune system has collapsed, or that it has a brain tumour, and either way death is imminent.
Abscesses around the roots of the teeth are a nuisance to treat, and are more painful than most ratty abscesses: pus can sometimes be expressed around the base of the tooth to relieve pressure, or a "head" may form in the skin under the animal's jaw and can be lanced. Abscesses behind the eyes, which often form at the same time as those on the teeth, are a very bad sign. Because of the inaccessible position they have to be treated with systemic (i.e. whole-system) antibiotics given by mouth or by injection, and in any case they generally indicate what my current vet calls "a hidden agenda:" - possibly a brain-tumour.
For dressing both abscesses and infected cuts I personally usually use antiseptic cream with a little tetracycline or oxytetracycline powder (from vet) mixed into it. Tea-tree oil also makes a good topical (i.e. applied directly to the infection rather than swallowed or injected) dressing, as does neat honey. A mixture of powdered turmeric and ginger has also been recommended, although I haven't tried it myself.
The most effective topical antiseptic of all is probably Dermisol, obtainable from your vet. However, I would use this only as a last resort - because it stings like blazes.