Note: These are the most common species and some are less common, grown indoors. More plants, poisonous for dogs and cats will be added to the list within the near future.
To keep on the safe side I have included plants that the ASPCA include on their toxic list. If you don't see a plant on the list above then it's most likely not known to be poisonous to the ASPCA.
There are many discussions about plants toxicity and how harmful they're, or how harmful they're not (it's suggested there are many that exaggerate). Toxic and poisonous just means an adverse reaction, which could be minor (i.e., mouth irritation) or extreme (rare).
It's worth noting, some of the unwanted affects (e.g, depression or anorexia) are from consuming amounts over a period of time. This can be avoided. If you see a pet chewing one of the plants then move it before your pet comes back to chew more and consumes too much. Check plant leaves for signs that leaves have been chewed on.
If a cat or dog consumes some of a plant and experiences mouth irritation then it's likely they wont return again (but we all know our beloved pets can be daft at times).
In my own experience we have never had a pet that has become sick from plants. My family have always had pets (cats and dogs) and plants without any problems.
The damn cats are the worst offenders, as you will know, if your a a cat owner. With dogs we can move the plants higher, but this does not stop cats. Any leaflet type plants, such as palms or dracaenas have been chewed the most, and again, cats were the worst offenders.
You can place cayenne pepper around the plant area or lemon/lime pieces or peel. Animals have a strong sense of smell and will not be tempted if they can smell pepper or citrus. Some of these will work, including a spray named bitter apple that many people have success with.
Another option worth trying. Provide a cat grass patch that could deter them from eating indoor plants. Certain grass types can also provide nutrition. See some ideas here at Amazon.
The best action to take is to call your local veterinary or one of the leading pet advisory services for your country (the US is ASPCA). The vet may advise you to just care for your pet while the negative effects clear, but it's worth a quick call to be sure your pet is not seriously sick.