Doormat is a mildly derogatory term for a type of submissive. Such a person (normally but not necessarily female) takes no responsibility, even as a submissive or a slave, in the relationship. Many are mistreated and put-upon. A doormat is typically not happy with their life, but will rarely do anything to change it due to low self-esteem and perceived lack of options. A doormat is not a BDSM role, but the term does crop up now and then in comparison.
From an external, vanilla perspective, the difference between a doormat and a sub or slave can be easily confused. Perhaps both are spanked when their partner wishes, made to do tasks they would rather not, restricted in their behaviour and perhaps even speech. Fundamentally, the difference is one of willingness and consent (at some point - consent would normally be a once-off event for a slave, though sometimes reaffirmed periodically). Look closer and the differences become very much more obvious: the doormat is normally unhappy, takes no encouragement as a person and lacks self-esteem. In contrast, the BDSM submissive is (on the whole, there being problems everywhere for everyone) lively, happy with her/his lot, proud of her/his partner and full (often very full) of confidence.
A BDSM submissive is respected and encouraged by her dominant to have self-respect. A doormat is often treated disrespectfully and is not encouraged to have any self-respect. A doormat is often open to real abuse and because of low self-esteem may continue to consent to that abuse, making her disrespect herself even more.