partition
Imagine measuring the area under a curve by slicing the base into thin vertical strips. Before you can build any approximating rectangles you must first decide where to make the cuts. A partition is exactly that list of cut points: it chops the interval into finitely many pieces so that the rest of the integration machinery has something to work on.
Formally, a partition P of a closed interval [a, b] is a finite set of points P = {x_0, x_1, ..., x_n} with a = x_0 < x_1 < ... < x_n = b. These points split [a, b] into n subintervals [x_(k-1), x_k]. The k-th subinterval has width delta_k = x_k - x_(k-1), and the widths sum to the whole length, delta_1 + ... + delta_n = b - a.
A partition by itself only records the geometry of the cuts; it says nothing about the function being integrated. To turn a partition into a number you still need a rule for choosing heights — either a sample point in each piece (giving a Riemann sum) or the supremum/infimum on each piece (giving Darboux sums). The points need not be evenly spaced; uneven partitions are not only allowed but essential for many proofs.
The uniform partition of [0, 1] into n equal pieces is P = {0, 1/n, 2/n, ..., (n-1)/n, 1}; here every width is delta_k = 1/n.
An equally spaced partition is the simplest case, but the definition allows any finite increasing list of cut points.
Some authors write a partition as the ordered tuple (x_0, ..., x_n) and some as the unordered set; the content is the same. Two partitions are usually compared by inclusion of their point sets, which is the key idea behind refinement.