Kant divides cognition into two - the Understanding & the Intuition. Both are required to make knowledge possible: The understanding supplies concepts which makes the objects represented by the intuition intelligible.
To make this clear - if I look at a chair without the Understanding all I see a mass of shapes and colours in space. It is pure sensation (intuition). By supplying the concept chair I make that pure sensation intelligible. On the other hand having the concept of chair without any sense-data is to render it contentless - it remains an abstraction.
The synthetic unity of consciousness is therefore an objective condition of all cognition, not merely something i myself need in order to cognize an object but rather something under which every intuition must stand in order to become an object for me, since in any other, and without this synthesis, the manifold would not be united in one consciousness
Judgements bring the concepts of the understanding and the objects of the intuition together. A judgement brings unity to several concepts & objects synthesised together: When I look at at the letter 'T' I can see that it is made of two straight lines but my judgement renders it as a unity as the letter 'T' - here I do not see its parts.
There are three parts to this generation: the manifold of parts (the lines), the synthesis (the process of make them into the letter 'T'), and the unity (the letter 'T' itself).
combination is the representation of the synthetic unity of the manifold
This is how concepts give unity to a manifold of intuition by synthesis.
It seems to me that this so called "synthetic unity of the manifold/apperception" is a reaction to David Hume's notion that the self is not a continuing entity, but rather a bundle of perceptions, which gives us the impression of a continuing self throughout existence.
Yes, that is roughly correct - Kant is reacting to Humes extreme skepticism by discovering the objective conditions (within ourselves) that makes knowledge of the world & ourself possible.