My experience is mainly in:

Yin yoga (focusing on increasing flexibility and range of movement)
Hatha yoga (here I focus on gaining strength and flexibility as well).
Lower back pain therapy (any pain related to stiffness, contracted muscles, and some neuropatic conditions like herniated disc and sciatic pain).
Restorative yoga (for calming the nervous system and reduce stress levels).

I combined all these styles to the needs of the students. In the first lesson I like to do an assesment of which are the parts of the body that need more attention, and also I take the time in this class to know what the student is expecting from the yoga practice.
Having a yoga mat is advisable but it really depends on the kind of postures we will do. If it is more with a yin / restorative approach the student does not really need it. For all my classes I use props like blocks, straps, bolster, chair. If you dont have them we can improvise with whatever you have available at home.