The mode of production includes the means of production which are human labor and the actual tools, machinery and capital that is used to produce goods. The mode of production also includes the relations of production which are the relations between people and the objects they use to make goods as well as the relations between the people themselves. In particular, these relations of production are the ways social classes interact and how workers function with employers, or how workers relate to those who own property and capital.
In Marxism, understanding the way goods are produced and consumed requires an understanding of how they are produced. These goods are first and foremost what we need to survive. So, the mode of production describes how we produce the necessities of our own survival. These goods also include things additional to food, shelter and clothing. This can include art, toys, and really any commodity. Anything that is produced is therefore, a part of our mode of sustaining life. Some of these goods are excesses or luxuries and profit is the excess value of production.
We don’t just produce goods. We produce our means of life which means that we reproduce our way of life. This includes our economic system but more fundamentally, our day to day activities. And Marx states that these day to day activities produce our ideas, politics, religion and cultural identities. In Marxism, the means of production are bound up with social interaction and culture. They are inseparable. In this context, culture is not just a concept of elite art. In this context, culture is broader. It describes shared attitudes and practices of a certain group, country or period of history. Everyday doings (modes of productions) inform the way we continue to perform those everyday doings and they inform the way we think about the world (culture).
In this context, culture emerges from the means of production. Or, as Marx said, consciousness emerges from life.
The mode of production is a concept created by Karl Marx. It is (like many of Marx's ideas) rather complicated. One way to try to state the idea in simple terms is that the mode of production is a combination of what we (in any particular country) produce and how we produce it.
To Marx, the mode of production that an economy uses pretty much defines that economy and, therefore, it defines the people who live in a given country. We are defined, for example, by whether we live and work in an economy where we do subsistence farming or in one where most people engage in white collar work that is done not with their hands, but with their brains. Marx would argue that these different kinds of modes of production make us into very different people, depending on which mode of production our society/economy uses.