Mindfulness as taught by the Buddha is more correctly called satipaṭṭhāna or mindfulness (sati) that is firmly established (paṭṭhāna). Nowadays it is often understood simply as awareness. But satipaṭṭhāna as a kind of awareness is not, as many Buddhists think, the general awareness normally associated with our day to day life.
The general awareness of our workaday life is usually only superficial and cursory. And even when it is deliberately developed to render it some depth and focus, it rarely approaches the depth and focus of satipaṭṭhāna mindfulness. It is usually engaged with a lot of worldly conceptual thoughts and ideas unlike satipaṭṭhāna mindfulness that, even though it usually begins with some conceptual objects like in-out breath or parts of the body, is finally aimed at the ultimate realities underlying those objects. The general awareness of daily life is also always directed towards solving the problems of our worldly mundane life unlike satipaṭṭhāna mindfulness that is aimed towards the arousing of transcendental spiritual insights that leads to the complete liberation from our one problem of existential suffering, the problem that the Buddha's teaching aims to solve.