At a basic level, current models of physics describe a force as the exchange of (virtual) bosons. This is just a model, but here's the general gist at what's a very basic level:
There are different types of fundamental forces: Strong, electroweak (electromagnetic + weak) and gravitational. On top of this, there is the Higgs Boson.
* The strong force is mediated by gluons
* The electromagnetic force is mediated by photons
* The weak force is mediated by the $W^+$, $W^-$ and $Z^0$ bosons
* gravitational force is mediated by the gravition, which has not actually been discovered, but is believed to exist
The Higgs, in this case, gives mass to the weak bosons by the process known as symmetry breaking.
'The standard model' in quantum field theory (QFT) describes how these (with the exception of the graviton) interact with each other and the other fundamental particles (fermions). Describing an 'object' as some sort of entangled wavefunction of the smaller objects in a larger Hilbert space:
To simplify the explanation, assume that there are only two objects in the universe which (to 'explain' the electromagnetic interaction) both have a net charge, with a negligible mass and are far enough apart that the other interactions have a negligible chance of occurring relative to having an electromagnetic interaction. Let's also assume that they are somehow observed at regular intervals (so that these objects actually have a defined position, albeit with some uncertainty) and that at some time, $t_0$, they are stationary.
What then happens is that these objects 'exchange a virtual photon' (or have some other valid QED interaction), which really means that the maths given by QED give the right solutions to the probabilities, strengths etc. of interactions (really, this is boiling down into physics is just a model - see the answer given by anna v...). What seems to be described by this maths is that one of the objects emits a photon with some momentum and so, by the 'law' of conservation of energy-momentum, experiences a change in momentum. This photon interacts with (gets absorbed by) the other object, which therefore also experiences a change in momentum. 800
In other words: the objects are now moving relative to each other! Having said all this, I need to stress: 'Exchanging virtual bosons' is really physics shorthand for 'the result you are looking for is given by performing the relevant mathematical perturbative expansion in QFT'. As this is difficult for everyday objects (read: practically impossible), we just use classical physics instead. Which, for everyday objects, is good enough.
The other interactions work by the same principle, but there are differences (strength of interaction, distance over which the interaction will realistically occur etc.) and relaxing any of the above assumptions means that other interactions also need to be included...
However, we have yet to include gravity. Suffice to say that the path (and hence, 'movement') of a particle is described by a world line, which, in the absence of other forces, is a geodesic
I've written all of the above as this is hopefully what you're looking for and so, will satisfy your curiosity, but what's important is:
1. The law of conservation of energy-momentum means that when two or more objects interact (so that they are not in equilibrium with each other), they experience a change in momentum, which is observed as either movement, change in mass, or some mixture of the two. Once an object has started moving relative to another object and it's no longer interacting with anything and its mass/energy is not changing, it stays at that velocity
2. All of the above is a model that fits an observation. Again, see the answer given by <PERSON>
3. Whether or not it's due to collapse of the wavefunction, decoherence or a number of other possibilities, is down to whichever interpretation of quantum mechanics you're using, which all give the same experimental results (at least, the valid ones do anyway) 343