All of these mean the same thing:
I'll be at my uncle's house in case you should need to reach me.
I'll be at my uncle's house in case you need to reach me.
I'll be at my uncle's house if you should need to reach me.
I'll be at my uncle's house if you need to reach me.
I'll be at my uncle's house should you need to reach me.
If something unpredictable happens, which causes you to need to reach me, you will be able to reach me at my uncle's house.
There is a chance that something will happen which will cause you to need to reach me. We can't anticipate exactly what that event will be, or even if it will happen. It is unlikely to happen, but if it does happen, it will just be one of those common things in life where a situation goes slightly wrong in an unexpected way. I will be at my uncle's house. So, if one of those vicissitudes happens, please call or visit me at my uncle's house, and the situation will turn out OK.
The central meanings of would and should
The central meaning of shall and should is to indicate the normal, proper, expected, desirable, good way that something happens—as opposed to will and would, which mean what actually “will” happen or might have happened. For example, “You should pay your bills promptly” is good advice. “You will pay your bills promptly” is a prediction about the future.
A central meaning shared by should and would is to indicate the consequence of a hypothetical condition, as opposed to shall and will, which primarily indicate propriety and the future tense (and volition). For example, “If you had invested in Apple 15 years ago, you would be wealthy today.”
In combination with its central meaning of normal, proper behavior or outcome, the conditional mood gives should an extra connotation of uncertainty. For example, “If you invest your money wisely, you should become wealthy.” That means the same as “If you invest your money wisely, and nothing strange and unfortunate happens, such as an economic collapse or you have a terrible accident or your bank gets robbed or who-knows-what, then you will become wealthy.” When you say should, this implies that something might go wrong, since unforeseen or undesirable events sometimes interfere with the normal flow of events.
Softening via the conditional mood
English, like many languages, uses the conditional mood to “soften” a statement. For example, people often say “I would like some coffee” to request coffee politely, because that’s softer than “I will have some coffee.” What makes it soft is that you’re using the conditional mood even though there’s no condition. You’re using the word that means a consequence of a condition, but there’s no condition.
This is what’s happening in your example. Notice that should is optional in your example. It just softens the idea that you might need to reach me. It suggests that you probably won’t need to reach me, or that you need not worry about it because even if it does happen, we have the situation well in hand.
You might wonder, if should’s central meaning is to indicate the normal, proper outcome, why, in the example, does it describe an unforeseeable situation where something goes wrong, perhaps an emergency? A full answer would take a long time to explain (see below), but here’s a useful short answer: this is one way the word should gets “stretched” to fulfill a wide variety of needs.
How to learn this
I recommend that you not try to memorize rules for when to use should, when to use would, etc. You should understand the main ideas and then just learn idioms through experience.
The main ideas are: (1) a central meaning of shall and should, not shared by will and would, is to indicate normal, proper action or outcome; (2) both should and would have a central meaning of indicating a consequence of a hypothetical condition; (3) these words get “stretched” into a wide variety of meanings—far beyond their central meanings.
The idioms are chaotic and highly varied. For example, you can use should as a synonym for if, especially when describing something going wrong. For example, “Should you miss your train, you can use my phone to call a cab.” Scientists study all these idioms and try to figure out exactly what all the rules are; and some scientists say that idioms don’t really follow rules. If you’re just learning how to communicate in English, you’re better off picking up the idioms through experience, slowly and patiently, than trying to memorize a definitive list of rules.
I haven’t told you the full extent of the messiness.
English often uses the past tense to express a hypothetical future situation (“If you left within the next hour, you should arrive by noon.”). Would is the past tense of will, and should is the past tense of shall even though shall is now rare and never put into the past tense! This is enough to make should suitable for some hypothetical future situations, though (introducing the condition rather than the consequence). For example, “If you should arrive late, the meeting will be canceled.”
I wrote a long explanation here. It's probably too long, but it explains some of the history of these words, how they overlap with each other, and how conflicts between them usually get resolved.