If partner opens one heart, what do I need to respond with one spade?
In Standard American bidding (and 2/1 game forcing for that matter), after partner's 1H opening, 1S promises at least 4 spades. After 1H-1S, partner's mostly likely calls are 1NT (5332 shape with no more than 14 HCP), 2 of a minor (5-4 or better in the bid suits, may have anything short of a game force in terms of values), 2H (6+ hearts, probably less than 17 high card points), 2NT (18-19 HCP, 5332 shape), or some number of spades.

After 1H-1S-1NT, your rebids:


2H: This is a tough one. Probably 5 spades, a doubleton heart honor, and shortness in one of the minors. You're taking the position that playing 2H in a 5-2 fit is better than playing 1NT (if you had 3 hearts and a weak hand, you shouldn't have bothered to show spades).
2S: The 2S rebid shows a weak hand, usually with 6+ spades (though it could be an exceptionally weak hand with shortness somewhere and only 5 spades, e.g. Qxxxx x Kxx xxxx, which rates to take 0-1 tricks at NT but a few tricks with spades as trumps).
2NT: Invitational values, likely exactly 4 spades and 2 hearts.
3NT: Game forcing values, likely exactly 4 spades and 2 hearts.
2m: You should have a discussion with your partners about this auction, but SAYC says this should be non-forcing and natural.
3H: 3-card heart support, invitational values.
pass: any weakish hand not covered above.


After 1H-1S-2m:


Pass: A weak hand with 3- or 4-card support for the minor and usually a singleton or void in hearts.
2H: A weak hand, virtually guaranteed to have a heart doubleton (again, with 3-card support, why not raise hearts in the first place?)
2S: Shows 6 spades or better (or, more rarely, a weak 5-5 in spades and the unbid minor with a singleton heart). With other shapes and a weak hand, correct to 2H with a doubleton heart or pass partner's minor.
2NT: Balanced, invitational.
3m: 4-card support for the minor, invitational values.
3S: 6+ spades, invitational values
The fourth suit at the cheapest level: forcing for one round, unable to make a natural bid at this point. Examples:


Interested in 3NT but unable to stop the unbid minor.
Interested in slam in one of partner's suits and thus unwilling to jump directly to game.
Interest in a spade slam but require more information from partner



After 1H-1S-2H:
 - 2S: This shows a weak hand with 6 spades or better and likely a heart void; why else correct from partner's 6-card suit to yours?


2NT: Natural, invitational.
3m: Natural and game forcing.
3H: Invitational with 3 hearts.


Obviously, this doesn't cover all the bases, but it lays out some of the most common situations.