Which day does "next Tuesday" refer to?

"next-Tuesday" would be better said as "Tuesday of next week". Thus, the answer to your question would be "on Tuesday".

My preferred use is to specify "this coming Tuesday" or "the Tuesday after next" when I talk about days, specifically because of the ambiguity of what "next Tuesday" could be.

That's just my preference, though.

This clearly has no definitive answer - usage varies between speakers. To me, "next Tuesday" means (strictly) the next instance of a Tuesday, although I just wouldn't use it on Sunday or Monday (preferring the day after tomorrow/tomorrow or an unqualified "Tuesday", which generally refers to the current week, past or present). However, I accept that this is personal and that others may sometimes mean "the next Tuesday but one", bizarre as that may seem.

This assumes that I know what day it is, which cannot always be relied on.

I know someone who would confuse it even further with the usage "Tuesday first" meaning the first Tuesday after today - where most people would interpret that as meaning a Tuesday which landed on the 1st of a month.

My answer to another question making a similar inquiry about using the phrase "next week"—

The issue isn't as definitive as you might think. Ultimately, it comes down to what interpretation of "next" you consider to be correct. For example, the American Heritage Dictionary (4th ed) defines 'next' simply as

Immediately following, as in time, order, or sequence

Following this definition, "next weekend" will always mean the weekend with the start date in closes proximity in time. If the phrase is used during a weekend, of course, you'd be referring to the weekend following the one you are currently experiencing.

However, the issue gets more complicated if you look to other definitions. The Oxford American Dictionary has a specific definition for 'next' when used in the context of time:

(of a day of the week) nearest (or the nearest but one) after the present : not this Wednesday, next Wednesday

Here, we're given the choice: it can either mean the weekend with the closest start date, or the following one (as specified by the parenthetical addition or the nearest but one).

This definition has come about mostly because of usage development. Many words and phrases in the english dictionary have meanings contrary to their technical definitions, and yet are still used commonly and considered valid. Thus, while technically "next" implies immediate sequence, it is used in other ways (which dictionaries like the OAD have accommodated for validity), so it's really a matter of personal preference. For example, it is unlikely that I will even use the phrase "next weekend" during the week, because some people might be confused as to what I am referring to. Instead, I will say "this weekend," unless it is currently the weekend, in which case I will say "next weekend."

There are several possibilities:

1. This Tuesday is the Tuesday of the week that we are in. Next Tuesday is the Tuesday of the week we'll be in after this one. This appears to be the most common usage, and makes most sense when you're trying to distinguish between imminent Tuesdays.
2. They both refer to the same date; this (coming) Tuesday, which is the next Tuesday on the calendar from today.
3. This Tuesday refers to this (coming) Tuesday, or the Tuesday that comes next on the calendar from today. Next Tuesday refers to the Tuesday after this (coming) Tuesday, or the Second Tuesday on the Calendar from today. This usage seems rare... but is actually what I understood for many years (although I was aware that people didn't always share my views).

Some people switch meanings on Tuesday. For example, they mean (1) on Sunday and Monday, but (2) from Wednesday onwards.

Occasionally, I've heard people talking who have different understanding of the terms, but who happen to come to the same conclusion about the date of an appointment!

And then there's "Last Tuesday"... but let's not get started on that one!

All in all, unless you know that you speak the same "calendar language" as someone else (and it isn't always obvious) your best bet is to:

1. Use a different way to express yourself, "Tuesday of Next Week", for example.
2. Use dates, "Tuesday 11th".
3. Or get out your calendar and point.

I think this question would be better if we spoke of Saturday rather than Tuesday because Saturday happens at (or near, depending on what day you feel starts the week off) the end of the week.

So on Monday the 1st I would say "this Saturday" to refer to the upcoming Saturday (the 6th). I would refer to "next Saturday" in contrast to "this Saturday". This and Next can't refer to the same thing, right? If you and I are catching taxis and I say "I'll take this taxi, you grab the next one" and you say "ok" and then try to climb into my taxi, arguing that this IS the next one... I'd regard you as insane.

The same applies to the days of the week and what I think is meant by "this [something]Day" is this week's [something]Day and "next" [something]Day as next week's something[Day].