I will collect here some things about tag-synonyms which seem basic to me. Let me know if I gave some incorrect information. And, of course, feel free to add another answer if you think some important piece of information is missing.
First of all, list of all tag synonyms can be found here: https://math.stackexchange.com/tags/synonyms
This lists includes tag synonyms that have already been approved and also tag synonyms which are only suggested. If you see the word "pending" in the list, then this tag synonym is not approved yet and it is possible to vote for or against the tag synonym.
The tag synonyms are not symmetric. Notice that column called "master" in the list of tag synonyms.
How can a tag-synonym be suggested? How is it approved
If you click on some existing tag, then you are shown the tag-info at the top of the page. Just below the tag-info you can click on the word "synonyms". You will get to the page where you see already existing tag synonyms and where you can suggest new ones. For example here. This page may look like this:
Here is another example, where you can see also the total score (=number of upvotes minus number of downvotes) for the particular tag synonym suggestion. (I am unaware of a possibility to see number of upvotes and downvotes separately. Both here and in the list of tag synonyms you only see total score for suggested tag synonyms.)
This is the place where you can vote (by click on the up/down arrow) for the tag synonym, which was suggested but no yet approve.
You can see there some relevant information.
- Only the users with more than 2500 reputation and a total answer score of 5 or more on the tag, can suggest tag synonyms.
- Only the users with score of 5 or more can vote for or against the tag synonym.
- Suggestions will be automatically approved when they reach a score of 4, and automatically deleted when they reach a score of -2.
- Moderators can create tag synonyms without need of getting votes from other users. (See also Mad Scientist's comment below.)
This process might be, in some cases, problematic. In particular for small tags there might be only a few users which can vote. So in practice, the tag synonyms are often discussed on meta. (There is a thread specifically for this purpose: Tag management 2015. However, if the synonym is likely to generate more extensive discussion, it is better to post a separate question.) If voting on meta indicates clear consensus, then moderators create the tag synonym. (In this case, votes from the users are not needed.)
Personally I think that discussion on meta has some advantages: You can see there also arguments for/against the particular suggestion, not only number indicating votes which have been cast on the tag synonym suggestion so far.