I am pretty new to playing D&D and our DM does not use figures with a grid, so we kind of use our imagination. According to the official D&D rules, are we doing it wrong?
You do not need to use miniatures and a battle grid.
In fact, playing on a grid at all is listed as a variant game option - it is not the default rules assumption.
No, you do not need to use miniatures
Miniatures and grids are not necessary for playing D&D 5e, they can be useful visual aids for combat as you can see exactly where you are relative to other characters but, outside of combat, there is little need to use them. Here is what the Dungeon Master’s Guide page 250 says:
In combat, players can often rely on your description to visualize where their characters are in relation to their surroundings and their enemies. Some complex battles, however, are easier to run with visual aids, the most common of which are miniatures and a grid. If you like to construct model terrain, build three-dimension dungeons, or draw maps on large vinyl mats, you should also consider using miniatures.
On the same page of the DMG, it also explicitly states that you do not have to use grids:
However, you don't have to use a grid a all. You can track distances with a tape measure, string craft sticks, or pipe cleaners cut to specific lengths.
Adjudicating Areas of Effect
If you wanted miniatures or tokens representing characters, such as dice or counters, but did not want grids, Xanathar’s Guide to Everything has a section on areas of effect on pages 86 - 88:
The template method uses two-dimensional shapes that represent different areas of effect. The aim of the method is to accurately portray the length and width of each area on the grid and to leave little doubt about which creatures are affected by it. You'll need to make these templates or find premade ones.
To use an area-of-effect template, apply it to the grid. If the terrain is flat, you can lay it on the surface; otherwise, hold the template above the surface and take note of which squares it covers or partially covers. [...] You can also use this method without a grid. If you do so , a creature is included in an area of effect if any part of the miniature's base is overlapped by the template.
Essentially, you just cut out circles, rectangles and triangles out of bits of paper or card and use those for adjudicating areas of effect. You’d probably want to label them with the sizes so you know what they represent but they could be useful.
From my Experience
As a DM, I don’t use miniatures or tactical maps (maps with grids on them) outside of combat.
Outside of combat, I find the use of miniatures unnecessary as you often don’t need to know the player’s exact location. As the DM describes the room, you have a good idea of where objects are relative to the room (the dresser is by the door, the bed is against the back wall and there is a balcony at the far side of the room). So, as you know where objects are relative to the room, when players interact with them, you also know where they are relative to the room. If combat starts or a trap is triggered, you know roughly where people are and who might be affected by the trap.
Instead of using tactical maps outside of combat, I use “line and rectangle” maps. As the name suggests, you only ever draw lines and rectangles, a rectangle represents a room and the line shows where the doors are and which rooms you can access from the room you are in. It looks visually similar to a flow chart. I find that this is enough to get an idea of what a dungeon looks like but it doesn't require miniatures or lots of preparation time. Also, as with a line and rectangle map you aren’t forced to draw with one-inch squares in mind, you can have large scale maps fit into a much smaller area.
In combat however, I find that visual aids are very useful. What I do is, when combat breaks out, the line and rectangle map goes to one side, revealing the battle mat. I quickly sketch the room or area that the players are in on the mat and any objects which might be relevant for combat, such as furniture, hazards, anything which might have a mechanical effect. For example, drawing the trunk of a tree which might provide cover or be used to gain elevation. If something is unlikely to have mechanics attached or be interacted with, such as the candle sticks on the fireplace or the paintings on the wall, I don’t draw them. I keep in mind they are there in case the player asks but they won’t be on the map.
So, your DM is not wrong to not use miniatures, every DM has a different way of playing the game. Many DM’s use tactical maps or dungeon tiles and miniatures all the time, others like myself only use tactical maps and miniatures when there is going to be combat and using other kinds of maps when there isn’t, some DM’s however prefer full “Theatre of the Mind” style games where grids and miniatures are not used at all.
While playing on a grid is very common, it is not necessary in any way; the 5e rules (Player's Handbook p192) make this clear - in fact playing with a grid is listed as a Variant rule (i.e. optional, at the DM's discretion). There are other variant rules that are very commonly used (multiclassing and feats are both optional rules for example, even though it seems to me the vast majority of groups regard them as standard); playing without those is 100% valid too.
I presume that since you ask the question, you don't have the Players Handbook readily available -- but you can still see for yourself: it's also in the Basic Rules on p74 (in the same form as in the book).
Note that some people play 5e using some grid/miniatures rules from earlier editions (like how to measure diagonal movement) in place of the rules in that "Variant" box in 5e. Again, this is completely valid, since the rules also make it clear that the DM is the final arbiter of the rules (and indeed positively encourages DMs to alter things to suit their setting, their play style and their group).
If you're all happy with how the game is going, there's nothing to do -- you'd be playing the game completely correctly in any case.
Some personal experience: I've played D&D on and off since the early 80s, in many campaigns, with many DMs. I have played a number of different versions of D&D (i) without any grid or minis, (ii) with a grid but no minis (e.g. on graph paper with dots and/or pencil initials marking positions and lines for movement, a bit like an old American football coaching blackboard), (iii) with minis but no grid; you can measure distances and just play on a table - though in a simpler form, we also just used groupings of minis to indicate groupings of combatants in a fight but leaving inter-group distances to the discretion of the DM, and (iv) with minis on a grid. [I've also played many other RPGs with or without a grid.] In some games we have switched between no-minis and minis.
In most versions of D&D I've played, I feel the use of some form of minis* and/or grid added to the game a little because it resolved problems where players and or DM had a different understanding of ranges and positions and so reduced a fair bit of "oh, if I'd realized he was over there, I wouldn't have moved where I did" and "But I was deliberately trying to position myself so he couldn't get both of us with a cone spell"-type discussions during combat.
On the other hand, some people feel it can interfere a bit with their immersion - they find it easier to let their imagination do its thing without them -- I don't feel that way, but I do understand it.
So while I prefer some level of representation of positioning to complete theater of the mind, many people are quite happy to play with no figures or grid and some quite prefer it. With or without it's all 100% valid D&D.
* here I intend the term in its broadest possible sense -- an unambiguous token or doodad (like a bottlecap or a die) for each creature is sufficient to count as a mini for the present discussion.
Not at all
The use of figures with D&D is a hold-over from a game Gary Gygax made before Dungeons and Dragons: Chainmail. Chainmail was a figures-oriented game, basically medieval combat using miniatures. When D&D was created, no rules for miniatures were included — but most of us who were playing in those early days had been playing Chainmail first — so it was natural to bring the two together.
The result was a tradition, not a rule. Figs are fun, but 99.9% of the D&D adventures I played did not use them.