The Morspoort is build in 1669 by architect Willem van der Helm (he also build the Zijlpoort). The current Morspoort replaced a wooden version build in 1640. The gate has a combination of old and new elements as possibly the architect wanted to fit the building better to it's surroundings. The name of the gate comes from the swamps that were found outside the city walls here ("morsch" means swamp).
Entering the city from this side (North West) must have been an unpleasant route, as here the gallows stood ready for executions. That's why the Rhine here is called Galgewater (translated literally; Gallow water).
Together with the Zijlpoort this is the only remaining city gate of Leiden.
The gate is a monument.
More info: http://erfgoed.leiden.nl/geschiedenis/monumenten/morspoort/ (Dutch)