Jesus most commonly uses the Greek word translated as "heaven", that is, ouranos, which means "sky." Biblical translation is very misleading in its translation of this word. For example, this word is almost always used with an article and most frequently in the plural, "the skies." Biblical translation usually edits out the article that comes before it and its plural nature.
The reason is clearly that the translators want us to think that Jesus was referring to our modern idea of "heaven," that is, the afterlife. However, there is very little in Jesus's words to support this idea. For example, this same word is translated as "sky", for example, when referring to the birds of the sky. English readers have no way of knowing that Jesus is always using the exact same word as "heaven" when referring to "sky."
The Plural and Singular
Jesus uses both singular and plural forms of this word, but the plural is much more common. For example, the phrase from the Lord's Prayer "who art in heaven" really says simple "the one in the skies." The word is usually used in the plural when referring to the sky as the location of the Father. The phrase translated as "the kingdom of heaven" is also almost always "the realm of the skies." See this article on the "kingdom of heaven" phrase.
However, in one verse, Matthew 7:21, Jesus uses both singular and plural together, singular for the "kingdom of the sky" and plural to describe the location of the Divine, "in the skies."
The Greek word for "sky" is also occasionally used in the form of an adjective. This is usually translated as "heavenly" in English, but the sense is the word "sky" as an adjective as we use it in "sky blue." Jesus uses this adjective to refer both the Father as the Sky Father and to the higher realm as the Sky Realm.
One possibility is that Jesus use the singular to refer to the earthly sky and the plural to refer to the larger universe. However, this is just speculation. When referring to the vault of the sky, the singular. "sky", is most often used.
Ouranos as a Metaphor
When we think about how Jesus's listeners would have heard this term, we have to realize that they would not have heard it as referring to the "heaven" that we know today, with clouds, angles, and the pearly gates. This ideas were not taken from Judaic traditions but Zoroastrian.
Nor was the sky seen as the place where spirits go in the afterlife. The Greek concept was the spirits go to the underworld, which is described by the Greek word hades, which is a word that Jesus uses that gets translated incorrectly as "hell." Jesus talked about the pleasant afterlife as the "bosom of Abraham" but he also describes a less pleasant afterlife as well but he doesn't use the word for either "heaven" or "hell."
For all people of Jesus's era, the "sky" or "skies" is a metaphor for higher things, the things that are not physical but conceptual and spiritual. The Greek concept was the purer things rose into the sky while the dross falls to earth. The highest ideals, the highest values, abstract ideas, and the things that are divine rather than earthly can all be thought of as part of the realm of the skies.
Jesus uses the plural "Father in the heavens" perhaps more often than he does the singular (Mat 6:1). The plural "heavens' works if we think of the word as meaning "that beyond earth." There are, after all, many places beyond earth. It also works with the concept of highest, since there are many different types of "height," at least conceptually. It works less well with the idea of "universe" unless we want to think that Christ was an early proponent of the multi-universe theories.
Heaven and Earth
This is the word used in the phrase "heaven and earth." Jesus never pairs "heaven" with "the world", that is, kosmos. Both words, kosmos and ouranos are masculine. If kosmos is the negative masculine of civilization, sky is the positive aspect of the masculine, a higher, more perfect plane, God-made, not man-made.
As an archetype in many cultures, "heaven" is the positive male aspect of nature paired with the positive female aspect of nature in the natural earth. The idea of the Sky Father follows with the broad cultural concept. Heaven and earth together are the "natural" reality, that is, what is created by God. Heaven and earth are, in a sense, the opposite of "the world" since the social word is what is man-made.
This same word, ouranos, also means a "chamber pot." So there is always something humorous about describing something as "the kingdom of heaven" since it also means "the kingdom of the piss pot." This meaning comes from the Greek word oureter, the source of our word ureter.
Another perhaps even common phrase using the term "heaven" is the "kingdom of heaven," which is discussed in this article along with related ideas.
Our Modern Idea of Heaven
The biggest problem is that when we hear the word "heaven" today, we have a set of images that have nothing to do with the way Christ meant the word. We see clouds, angels with wings, streets of gold, and pearly gates. Much of this is from childhood stories, and, unfortunately, at least some of it is borrowed from stories taken from other religions.
To get those images out of our heads, it may be best to think of "heaven" as simply "that which is beyond earth." However, this loses some of the sense of "great height" that we get from the Greek idea of "heaven" and up in the sky. This "height" is very important to Christ in a philosophical sense and for much of his wordplay. Heaven is the realm of the highest concepts. It is beyond earth both in the sense of being physically out of reach and in the sense of being higher in the sense of superior.
The Universe and Universal
Jesus often describes that Father as "in the skies," for example in Mat 5:45. We can read the as "in the universe", "beyond the earth," or "in the heights." Though often translated as "in heaven" in the English NT, Christ also describes the Father in Greek as "of heaven" or "heavenly" where "heaven" is used like an adjective. This can be translated as "the universal Father", "the Father beyond earth," or "the highest Father.
Though Jesus refers to the Father as "in" heaven, no one takes this to mean that the Father is physically contained within the universe or any group of universes. The sense is that the Father is active within the universe. "Inhabiting" the highest realms beyond earth.