"I now discuss different methods for forming matched pairs of treated and untreated subjects when matching on the propensity score. In doing so, several decisions must be made. First, one must choose between matching without replacement and matching with replacement (Rosenbaum, 2002). When using matching without replacement, once an untreated subject has been selected to be matched to a given treated subject, that untreated subject is no longer available for consideration as a potential match for subsequent treated subjects. As a result, each untreated subject is included in at most one matched set. In contrast, matching with replacement allows a given untreated subject to be included in more than one matched set. When matching with replacement is used, variance estimation must account for the fact that the same untreated subject may be in multiple matched sets (Hill & Reiter, 2006)."
So, this means ...
Matching with replacement: Comparison students will be matched with multiple treatment students.
Matching without replacement: One comparison student will be matched with one treatment student.
Also the definition of weights from R documentation:
A vector of length n that provides the weights assigned to each unit in the matching process. Unmatched units have weights equal to . Matched treated units have weight 1. Each matched control unit has weight proportional to the number of treatment units to which it was matched, and the sum of the control weights is equal to the number of uniquely matched control units.