I think the difference between the two types of examples that you've exhibited is the relative placement in time of the action in the "if" clause, and the action in the other clause.
- If it rains in the evening, we won't go for a walk - here, the event of raining occurs BEFORE the decision about whether to go for a walk.
- My teeth will rot if I eat too much sugar - presumably, I'll be eating the sugar BEFORE my teeth rot.
- If it will significantly increase complexity, don't implement this feature - here, the increasing of complexity occurs AFTER the implementation of the feature.
- I will give you money if it will make you happier - here, you becoming happier occurs AFTER I give you the money.
In all the cases where the "if" part happens first chronologically, we use the present tense. In the cases where the "if" part happens second, we use the future tense. However, because sentences of the first type are far more common than sentences in the second type, a good rule for learners to adopt is "don't use the future tense with IF".
To answer the official question, in every sentence (not every clause, but every sentence) the first verb in the main verb phrase must be one of
- a Present tense form (am, is, are, have, has, does, do, or
Verb + -Z₃, the 3SgPr inflection)
- a Past tense form (was, were, had, did, or
Verb + -ED, the Past inflection)
- a Modal auxiliary verb (can, could, may, might, will, would, shall, should, must)
English modal auxiliaries are not inflected for tense, so they are either not in any tense or they are always present tense, depending on what kind of theory of tense you're applying.
So either all English sentences are in a tense, or there are some that aren't. But that's just how one describes the language -- it doesn't affect the grammar.
Time in English is frequently indicated by tense, but often enough it's also -- or even only -- a matter of the words or constructions used. See the Deixis Lectures for more on expressions of Time.
English tensed verb forms, however, often specifically don't refer to time. For instance, the Present tense, when used with an active verb, is most likely to refer to an habitual occurrence than to the present time. E.g,
- Bill walks to school means he walks (almost) every time he goes to school.
- That dog bites means that the dog has been known to bite people on some occasion(s).
- Mary drives a Toyota means that Mary usually drives (and probably owns) a Toyota.
None of these refer to what Bill, the dog, or Mary are doing at the present time -- neither the time of speaking nor the "present" of a narrative. This is called a generic construction.
The particular use in the original question licenses the use of a past verb form to indicate an unreal supposition, much the way certain regular subjunctive verb endings do in European languages; but only sporadically -- not regularly. This counterfactual conditional construction, like most archaic remnants, is idiomatic, and governed by only a few constructions and verbs. So one finds
- I wish I were home now.
- If I were you, ...
- If I had the money now, I'd give it to you.
Best Answer
Traditionally, it is considered neither, though it is sometimes called a mood for lack of a better word. The word function would seem the best term. I am assuming that you are talking about the word would in your example. The adjective "conditional" just means "related to a condition"; when used as a noun, it usually refers to a function of the past subjunctive tense of modal verbs (would, could, might, and should).
There are three traditional moods in English, indicative, subjunctive, and imperative; there are several tenses, which are combinations of moods, aspects, and temporal properties (you might say past and present are aspects, though they are usually not so called).
There are three types of conditional sentences, as Henry has pointed out, which are mainly just sentences in which an explicit or implicit condition is present. There is a function of the past subjunctive that is called conditional because it is used with one of these types, the so-called hypothetical condition (if he were rich, he would be unhappy). I think that is the one you mean; while it is sometimes called a mood, I find this unclear and confusing: how can something be of two moods, both subjunctive and conditional? Function is the term that both fits best with established terminology and best describes what kind of phenomenon it is.