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In general, the following setup is used in a titration (there may be some exceptions):
During a titration, we are generally performing a reaction using a substance/solution of known concentration (i.e. our titrant) with a substance whose concentration is not known (i.e. our analyte, the substance we are "analysing"). We are looking to work out this concentration based on the amounts of each we use in allowing all of our analyte to completely react and form products.
So, we take a fixed volume of our analyte and place this into an Erlenmeyer (or conical) flask, below the burette. This will contain our titrant (such as a strong acid or base), which we add to the analyte until our reaction has reached completion, usually based on the presence of an indicator added to the concical flask or a natural colour change after all of the analyte molecules have reacted appropriately with the number/volume of titrant molecules we have added.
For accuracy, a appropriately sized pipette would be used to measure out our fixed volume of analyte into the Erlenmeyer flask as opposed to a graduated cylinder, which can also measure volumes of solution. This is due to the superior accuracy of the pipette in delivering a fixed volume/number of moles of the substance, which is important in being able to calculate its concentration accurately as part of the titration.
I remember hearing of some instances that in acid-base titrations (where you are performing a neutralisation reaction) it may be better to place the the base in the conical flask, as certain strong bases can lead to blockages in the taps of burettes if they are not cleaned our properly after use. I can't remember coming across any instances where that would mean the position of the titrant and analyte being swapped around though!