The notion of Being-in-the-World is fundamental to Heidegger's phenomenological investigation into the being of Dasein given that Dasein simply is the way of being-in-the-world. What Heidegger is getting at by this compound expression is detailed across three large chapters. Critically, he stresses the equiprimordiality of its "structural moments" - for Heidegger it is crucial not to crudely analyse being-in-the-world in the sense of breaking it down into parts and trying to find some homogenous ground through which it all makes sense. Being-in-the-world is the ground and it is a whole phenomenon.
Now, he does not deny that its structural moments can be appreciated by themselves (the world, being-in, and the "who" of being-in-the-world) otherwise his analysis could not get off the ground. He simply insists that phenomenologically, i.e. in reality, the phenomenon does not present itself as a sum of isolated parts. This holistic character is reflected in the analysis itself which cannot be strictly organised into chapters dealing exclusively with one of the three structural moments. The chapter dealing with world, for example, must also consider being-in, if only briefly, in order to make sense.
So what does Heidegger mean by being-in-the-world? We'll begin as Heidegger does by offering "knowing" as a founded mode of being-in. Knowing is a form which our intentionality can take in its being directed towards the world in some way. Being-in-the-world in a knowing manner simply entails having knowledge about something in the world. The typical view which Heidegger is seeking to overthrow here is that knowledge necessarily precedes all modes of being-in. He wants us to see that there are more original ways of being-in - one can love something, or use something, or produce something, be frightened by something. Heidegger's analysis will show us that knowing depends upon these more original modes of being-in. What is known is always something which is encountered in pre-cognitive activity. That you love Earl Grey, or that your friend studies philosophy - in order to be able to know these things as propositions there must first be a thing which is known, an original encounter. How we understand things, as we'll see, is through concernful lived activity. If Heidegger is right then it simply makes no sense to put knowing before living.
The world, then, is that towards which modes of being-in are directed. The being [sein] of beings [seindes] within the world includes Dasein, the ready-to-hand, and the present-at-hand. Our understanding of these three enables those beings to demonstrate themselves, to be intelligible. Understanding the being of the ready-to-hand, for instance, involves always already grasping how something can be for something, as the keyboard is for typing or the mug is for drinking from. However, the keyboard and mug considered from a disinterested perspective are merely objectively present things occuring at a place in space at a given time, divorced from the practical context in which they make sense (have meaning) as what they are. This disinterested perspective reveals things as they are under the aspect of the present-at-hand.
But what Heidegger wants to show us is that things are never initially encountered as present-at-hand, i.e. as meaningless objects simply "there" at a point in space and time which then have value-predicates foisted on them by a subject (see previous for a more in-depth set of notes on Heidegger's reinterpretation of the present-at-hand with regard to its grounding in the engaged/lived/pratical context). Things are initially encountered by us in some sort of meaningful practical context. As we said above, they must be able to touch us in some way, to be significant in order to be understood at all. How, then, does this understanding come to pass?
In order to appreciate this more fully consider the following example: walking through a busy Christmas market you see a child aimlessly striding towards you with some candy floss. Without thought you immediately perceive the child as threatening the sugarless and unsticky status of your clothes and step aside to avoid getting the candy floss on you. How is it that the candy floss was understood with respect to its being sticky? A typical account might insist that at some point we identified the fact that candy floss is sticky. Inferring from the stickiness of candy floss and the desire not to be sticky we arrive at the resolution to step aside. Sounds plausible, but what this account misses is the original phenomenon in which the candy floss was first of all able to demonstrate itself - the phenomenon of being-in-the-world.
How does the candy floss come to be understood as "sticky"? It is not that stickiness is directly intelligible by any means! In fact, taken outside of this lived context the candy floss is merely an object with no other meaning (even its being identified as an object depends upon this meaningful context in which it is initially encountered). How, then, does it demonstrate its nature? Our concernful engagement allowed the stickiness of the candy floss to come forward and reveal itself as a threat to our otherwise clean and dry clothes. Through its being relevant to our concern it is able to demonstrate its significance - that it is sticky.
That we were averse to getting candy floss on ourselves results from, as mentioned, being concerned about our clothes. In wanting our clothes to stay clean and dry various things which would make our clothes dirty show up for us in their relevance to this particular concern. Being concerned in this manner signifies various courses of action (avoidance, for example). We already had to know that candy floss was sticky, but how we came to know this originally was still in concerned engagement with it - not disinterested knowing. If we were not at all concerned about things "getting sticky" the stickiness of candy floss would never have become salient and we could never have known about it.
Within this context of involvement the things of the world are able to be understood as what they are - as heavy, or difficult, or indeed "sticky". Knowledge itself is organised with respect to a concernful perspective. "Pure" knowledge is a non-concept. How could we call the performance "delightful" if we are not captured by it? Or "boring" if it fails to affect us? How could we call a shattered mug "broken" if it isn't already understood as for drinking from? Being "broken" implies being in a state such that it can no longer do what it is supposed to. This "supposed to" derives its sense from the phenomenon of world as an in-order-to. An in-order-to (e.g. in-order-to drink tea) denotes a with-which (the mug) and a towards-which (drinking tea). Practical involvements reveal the nature of the things engaged with. The world, in Heidegger's sense, just is this interrelated nexus of people, things, and purposes. It is what Heidegger calls a "clearing", an open space in which things can be understood as what they are. Being-in-the-world is the practical/engaged/lived activity in which Dasein secures its understanding of the things in the world (not just propositional knowing-that but engaged knowing-how).
With this cursory overview we can more easily appreciate "being-in" and "the world" as equiprimordial structural moments of being-in-the-world. Without things and such in the world for Dasein to make use of or be threatened or seduced by Dasein can't make use of or be threatened or seduced by anything! But similarly, without Dasein these things could never enter into a lived context and demonstrate what they are by virtue of their relevance to our concerns. Beings press in upon us as we press in upon them.