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When I'm modelling, I often end up subdividing a lot of edges and creating a lot of faces/edges. However, I often end up having a face that doesn't follow the edges or vertices assigned to it when it's created. 
Can I get any help with what this is? or how to solve it? It would help a lot. Thanks :)
In short, the face shown in your screenshot has 2 problems:
Chances are that you'd like to get rid of faces of this type and make sure they won't come up in future.
The thing is that there probably aren't any ways to avoid faces of any type in an automated way. It could happen that object is allowed to have Ngons on planning / modeling stage, but isn't on the final stage of working on it (when lighting / rendering). So no one knows if you are designing model or cleaning it. See When should N-Gons be used, and when shouldn't they?
The only viable option is to keep track of your topology which may be quite important for overall look of your model. So if it's a problem to have these faces, make sure not to create them! Do not subdivide faces while modeling (unless subdividing all of them); use LoopCut tool to add details along with managing edge flow orientation.
See also Topology Guides blog for great tips about modeling.
A lot could be written about different ways of modeling while preserving topology as good as possible, but generally 2 global principles should be listed:
cleaning up the model in the end of modeling stage (this could include retopologizing too);
modeling with quads or quads-tris only from the beginning.
While second comes mostly with practice, the first could be achieved with some tools which already exist (keep in mind that these tools are meant to be used for finding Ngons in meshes and finding / repairing non-planar faces in particular).
See also:
How to flatten a non-planar face to a targeted plane?
How to make all quads or ngons on your mesh planar (2d)?
How to flatten a face to avoid distortion (make an ngon planar)
In Edit mode, Ctrl + Tab, choose Vertices, there would be a vertex where the pointy face is, select that vertex, then X, choose Dissolve vertices, this will remove this vertex non distuctively.
The reason you are getting these artifacts is probably hitting a button by mistake at some stage, keep in mind that some operations are multilevel, as extruding by hitting E for example, right clicking before confirming would look like cancels the operation, yet it only cancels moving the selected sub-object, but kept the copy, you need to undo in this case, several other operations act the same.
