The Subjunctive is, basically, when you speak of actions that are not real but rather desired, asked to be performed or just actions that might have happened.
One of the important uses of conditional (a.k.a. subjunctive) in Russian is with «чтобы» ("in order to") to express the idea of some action being required or asked for from someone.
1 entity→infinitive. The sentence is pretty straightforward when you only have one person:
2 entities→past. When A does something for B to do something, use PAST tense in subjunctive:
In more formal English it would look like "Mother asks that you be home tomorrow", though English has more idiomatic ways of saying that.
So, use the past to form the subjunctive in any structures like "A told B to do something", "A did X so that B would do Y", "We need that A do X" etc. The analogy with the English "that" (which might come off as overly formal) is probably a good way to grasp all these structures, since, unfortunately, in more idiomatic English the sentence structures would be quite different to the Russian.
◉ Pay attention to the use of aspect. When asking someone NOT to do something, imperfective is normally used.
The conditional, unsurprisingly, is also used in conditional sentences. When you describe hypothetical (unreal) situations, you always use Past + the particle бы. This particle normally comes right after если or after the subject / the verb:
Russian does not distinguish (grammatically) between "would be" and "would have been"; they both just use past and are distinguished based on what makes sense in a given situation.