A merge is what happens when two existing TV Tropes pages are actually be describing the same concept, but have different names and appear to have been developed as pages in ignorance of one another. So as to avoid redundancy, the content of these pages is merged into a single page that uses various parts of both pages' descriptions and all relevant examples. A split is the opposite of this- when a a page is seen as incredibly broad, it is divided into multiple more specific sections that deal with more precise variations of the same general concept.
This terminology is some of the oldest on the wiki, even predating the YKTTW phase of trope renames- my first major edit was a split of a page I felt had grown unreasonably long. These terms are the practical actions related to the Lumper Vs Splitter distinction- back when the wiki was a smaller environment, users were encouraged to either think of themselves as Lumpers (lump similar ideas on the same trope page) or Splitters (split similar ideas into their own trope pages) in regards to the tropes they wished to develop. Merges and splits are corrections made to errant thinking in the YKTTW process that resulted in the development of pages that were, in various ways, flawed.
Merges and splits were very odd page actions as the YKTTW trope rename process pushed onward because from any objective standpoint, merges and splits were just as if not much more important than renames, but because their purpose was seen as objectively organizational (compared to the very subjective opinions that made up a rename debate), they were usually ignored. Merges in particular are quite a bit like renames in everything save for this general motive. An individual simply opens an edit box for the page to be merged, reintegrates the text into the final article, and slaps a redirect into the formatting so that anyone attempting to reach this old page is now sent into its new, fully merged counterpart. This may have been the original purpose of the redirect command, though it's just as likely that redirects were part of the wiki platform from the beginning and this was just an obvious way to use them when these problems first started cropping up.
Redirects were greatly preferred to the alternate method of merging pages of manually changing every page's use of the obsolete Wiki Word. In addition to greatly reducing the work involved in merging pages, the redirect was just seen as a polite. A troper could be perplexed or confused to discover one day that all of the links to this one trope had been replaced by links to some other page. While this often abated once said troper looked at the page history or discussion to realize what happened, tropers are a paranoid lot and may still be alarmed into believing that they have suddenly drifted into an alternate reality.
Renamed tropes featured the same general issue of linking. Redirect or not, people were confused to click on one familiar trope title only to suddenly find themselves on a page with a completely different name. For this reason, it became common courtesy to change some (usually half) of the Wiki Words to the new title and letting the other half remain the old redirected title. As with merges, this gave the impression of gradual integration- tropers did not need to adjust right away, but they needed to note that the page was different now and any future Wiki Words should use the proper title.
Splits were a far less complicated affair. These usually ended up in YKTTW, as YKTTW's specialty was creating new tropes. Less exciting splits, such as creating alternate namespaces to house exceptionally large sections of examples, have traditionally been performed informally by whosoever was passing by and thought a split would make for better organized pages. Individuals like me, for the most part.
In any event, the existence of the Trope Rename Forum as the source of legitimate renames eventually led to the Merges and Splits forum being created to be the source of legitimate merges and splits. This forum was not an especially popular one. I can only ever recall actually discussing one page there, and soon found myself annoyed that nobody had any interest in a split, and yet at the same time I couldn't exactly propose a rename without confusing the heck out of everyone because these functions belonged to two different forums. The forum thread I posted as to this issue is what led to the creation of Trope Repair Shop- the institution that currently manages the wiki's quality control.