'Singer/songwriter' refers to the genre in which musicians compose, write and sing their own music (both solo and as a group)- none of the commissioning that has become practically all of current 'popular music'. Though not exclusive to, it's closely linked to folk-acoustic music.
Pianos and guitars are very common in this genre, though it can encompass whatever instruments (or technology) the musician chooses. It's also important to note that the majority- if not all- songs classed as singer/songwriter will belong to at least one other genre. It could be said that the term is less of a music style, and more about the composition.
Songs in this genre tend to be personal to the musician, and are often used as vehicles for political or social views.
I could easily name a lot of singer-songwriter songs (most of my favourite musicians fall into this category!)- however, I'm trying to expand my musical horizons, so I want to find something a bit different... (BB, I thought of you when I chucked the last one in- first time I've listened to it, it's great!)
O Children- Nick Cave And The Bad Seeds
Skinny Love- Bon Iver
How To Be A Heartbreaker- Marina And The Diamonds
A New England- Billy Bragg