What do Buddhists think in quick prayers (at graves, near altars etc.)?

When declared Buddhists and Eastern Asians in general quickly pray next to graves and altars in silence, what kinds of thoughts go on their minds? Of course, it's not possible to read people's minds, but how are they taught to pray in such occasions and places? I'm curious especially about non-practicing Buddhists who aren't very religious in general and may not know things like mantras by heart, but still do that.