der Raum
room
A limited, usually enclosed area inside a building, such as a room or a hall.
Die Wohnung hat vier Räume, eine Küche und ein Bad.
The apartment has four rooms, a kitchen, and a bathroom.

room / space
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room
A limited, usually enclosed area inside a building, such as a room or a hall.
Die Wohnung hat vier Räume, eine Küche und ein Bad.
The apartment has four rooms, a kitchen, and a bathroom.

'Raum' (room/space) is the basis for many useful compound nouns. Examples include 'Wohnraum' (living space), 'Stauraum' (storage space), or 'Weltraum' (outer space/the universe).
Unter dem Bett gibt es viel Stauraum für Kisten.
There is a lot of storage space for boxes under the bed.
Pay attention to the difference between accusative and dative with the preposition 'in'. 'In den Raum gehen' (to go into the room, accusative) describes movement, while 'im Raum sein' (to be in the room, dative) describes a location.
Bitte bringen Sie die Stühle in den Besprechungsraum.
Please bring the chairs into the meeting room.
In the abstract meaning of 'space' or 'opportunity', 'Raum' is often combined with verbs such as 'geben' (to give), 'lassen' (to leave/allow), 'schaffen' (to create), or 'bieten' (to offer). It is usually followed by the preposition 'für' (for).
Ein guter Chef gibt seinem Team Raum für eigene Ideen.
A good boss gives their team space for their own ideas.
When 'Raum' is used with abstract nouns such as 'Kreativität' (creativity), 'Ideen' (ideas), 'Gefühle' (feelings), or 'Entwicklung' (development), it almost always has the figurative meaning of 'space' or 'opportunity', not a physical room.
In dieser Beziehung war kein Raum für persönliche Entwicklung.
In this relationship, there was no space for personal development.
In geographic contexts, 'Raum' is often combined with an adjective to describe a region. Typical examples are 'der ländliche Raum' (rural areas) or 'der europäische Raum' (the European area/region).
Die Arbeitslosigkeit ist im ländlichen Raum höher als in den Städten.
Unemployment is higher in rural areas than in cities.
place / space
'Platz' and 'Raum' can both mean that there is enough free space available. 'Platz' is used very often in everyday language when asking whether something fits inside or whether someone can sit or stand somewhere. 'Raum' means either a room or, more generally, an area where something is possible. For that reason, 'Raum' often sounds a bit more general or formal, except when an actual room is meant. Learners confuse the words because both have to do with free space. If you mean 'does it still fit in there?', 'Platz' is usually more natural. If you mean a room or a larger area, use 'Raum'.

Für den Kurs brauchen wir einen hellen Raum mit großen Tischen.
For the course, we need a bright room with large tables.
Im Koffer ist noch genug Platz für eine Jacke.
There is still enough space in the suitcase for a jacket.