How are landing minimums decided for a flight? Does the captain make the decision depending on weather conditions or is it preset for each airport?
The terms you are looking for are Decision Altitude (DA) and Decision Height (DH). If no visual reference to the runway is established at the DA or DH, a missed approach has to be initiated. So based on the weather and the certification of the airport (CAT I / CAT II / CAT III), the pilot will make the decision to land or not land at the DA/DH.
(Image Source: SKYbrary)
Decision altitude (DA) is referenced to mean sea level and decision height (DH) is referenced to the threshold elevation.
The DH for Category II and III approaches is invariably assessed by reference to a radio altimeter and never a barometric altimeter; therefore the minima can only be expressed as DH and not DA. For approaches with DH of 200ft or higher, radio altimeter reading would be unreliable due to the unevenness of the terrain; therfore a barometric altimeter is always used and the minima may be expressed as DH or DA.
The required visual reference means that section of the visual aids or of the approach area which should have been in view for sufficient time for the pilot to have made an assessment of the aircraft position and rate of change of position, in relation to the desired flight path. In Category III operations, the required visual reference is that specified for the particular procedure and operation.
For convenience where both expressions are used they may be written in the form "decision altitude/height" and abbreviated "DA/H"
CAT II ILS Approach LGTS
Is it legal to land in bad visibility conditions without ILS?
Are zero visibility landings practical and desirable for commercial aviation?
How can landing in zero visibility be safe?
It is different and set at every airport, at each runway, for each category (size) of plane, and for each procedure that is followed. For example a VOR approach has higher minimums than a Localizer and then again than an ILS because of the precision of the approach.
In general almost every approach to any IFR runway has different minimums including if it's a straight in (lower) or a circle to land (higher).
There are also planes with radar ground sensing that can fly what are called CAT III approaches and can land in almost zero-zero weather. Those are highly automated and computer controlled landing systems.