To directly answer the question, there is one category of words which require a definite article ("der", "die", "das", "dem", "den", "des").
Words only with definite article
These words are considered to denote a concrete (unambiguous) concept of something, i.e. one single indivisible (statistical) population or generality, or otherwise things which conceptually only denote whole pieces of something. The exact words seem to be special cases because whether a word conveys this concept is specific to the German language.
Respective words cannot go without article, such as "die Schule" (school), "das Atom" and the mentioned outer space ("der Weltraum", "das Weltall") from the comment which refers to the single entirety of physical space outside of our world, it can be literally translated as "space of world".
It is not a strict rule though. At least colloquially, you can omit the definite article for some of the words such as school or apprenticeship without semantic change.
Words typically used without article
There is a special class of words that usually is not used with articles: the names of non-physical, cognitive and psychological things, e.g. names of living beings and people (souls), school subjects, personified names of objects with a personal relationship (such as vehicles or devices).
Articles are rare for person names whereas the other things can usually be mentioned in its generality, such as "die Chemie", the chemistry, "das Deutsche", everything German, but we often do not interact with the generality as a whole and therefore articles are omitted, e.g. "Ich lerne Deutsch.", I am learning German.
Therefore, using articles with person names can have a derogative sound (particularly in combination with only a surname), and indefinite articles with people names even can have an animalistic sound. Definite Articles also can be use in a respectful sense to make someone sound special and unique by using a definite article, particularly in combination with a forename or fullname.
In the contrary, the absence of an article does not apply for names of places, epoches, objects or events.
Words which can be used with and without article
The article can be even critical to distinguish terms because the same term can mean different things when used with or without article, such as "Stuhl" or school subjects (which are basically names that are not used with articles) vs. the general science.
Words without article typically can also be used with article, at least when there is match with the conversation's context.
For example, "Wasser", water, which can be used without article but you may also say "Das Wasser reicht mir bis zum Hals!" (the water reaches up to my throat). (This means, the specific water concerning the speaker in the situation.) Also possible: "Ein Wasser, bitte!" (one water please) at a bar. (Of course, it means one bottle or glass of water as it is offered on the menu.) One could also say "Wasser, bitte!" but it does not explicitly refer to one specific unit of it.
When definite articles are used, they semantically imply specificity, or refer to the specific entire generality of something which is possibly the most concrete thing if no context can be matched.
The example "Die Schuhe" (the shoes) implies either specific shoes or a specific set of shoes as known in the context or the shoes in its entire generality (which is less common), e.g. in "Der Schuh begleitet den Menschen seit Jahrtausenden." ("den Menschen" (singular) or "die Menschen" (plural) also refers to the concrete generality of humankind).
For words which can be used with and without definite article, the removal of the definite article expresses different semantics:
"Simon vermisst seine Schuhe. Lina hat die Schuhe!" (Lina has the shoes, refering to Simon's contextual shoes that he is missing.)
"Simon vermisst seine Schuhe. Line hat Schuhe!" (Simon misses his shoes. But Lina has shoes and her shoes do not need to be related to Simon's shoes in the context.)
In some non-contextual special cases, there might be no practical difference whether a definite article is used or not. This happens in combination with "gleich" (equal). "Sie haben die gleichen Schuhe!" and "Sie haben gleiche Schuhe!" (they/you have equal shoes).
Even though, there can be a difference in meaning. The usage of a definite article in "die gleichen Schuhe" could express some proximity of the shoes to the speaker. The speaker compares specific shoes, probably his own with the shoes of someone else whereas the absence of the definite article in the second case could express distance to the speaker, potentially implying that other two people have mutually equal shoes.
But still, both sentences could be interpreted in the same way.
This independence of an article seems to be a special case because it does not work with "identisch" (identical). One can only say "Sie haben die identischen Schuhe!" when it refers to context.
When it comes to learning German, it could make sense to remember that the definite article always indicates a concrete (unambiguous) thing, something which is conceptually indivisible, exists as a whole or in whole pieces or specifically refers to the context. In the English language, these are similar to the "countable" concepts (for which we can use "many") as opposed to uncountable or subdividable things (for which we use "much") which typically allow for omission of the article.
Absence of a definite articles refers to an abstract amount of something that can be divided or shared, like food, money, subjects, or to the names of non-physical mental things, particularly souls, even though they can also be used with article.
But eventually, for me, it does not seem as if there is a consistent and complete logic system. It has special cases which need to be learnt by heart.