The word "excise" literally means "a piece cut off" (Latin _excido_). Excise duties were a portion set aside for the government of the value of items produced and consumed within the country. This distinguishes them from customs duties placed on items crossing the border, either for import or export. The OED notes, however, that until the 20th century this distinction was not strictly followed, so that "excise" duties were also placed on some items of foreign manufacture.
This type of taxation was new in Pepys's time, having first been created by Parliament to help pay for the Civil War, in imitation of a practice already established in Holland. The Royalists for their part quickly followed suit. The OED contains a number of highly interesting example sentences straight from the early days of the excise; note particularly how unpopular these duties were from the very beginning:
1642 Decl. Ho. Com. 8 Oct., Aspersions are by malignant persons cast upon this House that they intend to . . lay excizes upon . . commodities. 1643 Ord. Lords & Com. 22 July sec. 2 An Office . . is hereby erected . . called or known by the name of the Office of Excise or New Impost. 1647 Clarendon Hist. Reb. VII. (1843) 471/1 This [July 22, 1643] was the first time that ever the name of the payment of excise was heard of, or practised in England. 1667 Marvell To a Painter, Excise . . With hundred rows of teeth, the shark exceeds, And on all trades like Cassawar she feeds.
Among the first excise duties passed was that on beer, again following the Dutch precedent. After the Restoration the excise became more important than ever, because such duties took the place of the old feudal duties that until that time had been used to maintain the army.
[This entry expands on a previous one about the excise on beer and its long-term effect on changing England's drinking habits. That entry can be found at http://www.pepysdiary.com/encyclopedia/308/#c2526 .]