When partner bids a suit after an opponent has already opened the bidding, you should treat the bid in many ways as an opening bid.  But remember the key differences:

1. Partner is guaranteeing a 5-card suit.  (An opening bid may be based on only a 4-card suit.)
2. Partner may not have as many points.  (An opening bid promises 12 HCP, but partner might overcall at the 1-level on as little as 8 HCP.)

The upshot of this is that:

1. You only need to hold three cards in partners suit to be content to play with that suit as trumps.
2. You need to hold more points to consider looking for game contracts, in particular if you don't have a good fit.

Remember that you should be satisfied with partner's choice of suit even if you only hold three cards.  It is seldom right to introduce your own long suit if you are happy with partner's choice.

If you do like partner's suit, then you should bid as if partner had opened that suit.  (Although partner might not have many points, the quality of the suit should compensate).

For example, suppose partner overcalls 1♠, and you hold three spades yourself, then you should let him know that you like spades (by bidding spades as well), and show how many points you have by the level at which you bid:

• 0 - 5 HCP - just pass!
• 6 - 9 HCP - bid 2♠.
• 10 - 12 HCP - bid 3♠

Had partner opened 1♥, you would have responded 3♥ to show that you like hearts, and that you have a quite good hand (10-12 HCPs) - not quite enough points for game, but you don't need much extra from partner.

When partner overcalls 1♥ after an opponent opens the bidding, you respond in exactly the same way.  Partner has a little extra, and so accepts the invitation, bidding 4♥.

Note that you don't have 25 HCP between the two hands, but the extra 'shape' and quality of the trump suit compensates for the lack of high cards, and makes the game contract a good prospect.

You should be satisfied with partner's suit even if you only have 3 cards.  But if you have just two cards in partner's suit, then might want to consider bidding a suit of your own, or even trying No Trumps.

However, this is when you need to remember that partner might have fewer points than if he'd opened the bidding.  For example, a 1-level overcall shows about 8-15 HCP, whereas an opening would have shown 12-19.
Remember also that because partner has at least 5 cards in his own suit, it's less likely that he will have many cards in your own suit.

Therefore...

To respond to an opening bid of 1 of a suit, you would need about 6 HCP or more.  But to respond to an overcall you need at least 9 HCP.

To bid a new suit, you need a good suit - at least a 5-card suit.
To bid No Trumps, you have to have a 'stopper' in your opponent's suit.

When partner overcalls 1♠, you should think that there is a possibility that there is a game contract available, but you don't want to play in spades!  So try bidding your own suit, and see if you can negotiate a better contract.

After you bid your own best suit, partner surprises you by bidding 2♥.  This is good news! You know now that you have an 8-card fit in that suit, and even though partner might not have many High Card Points, you have enough shape and strength to bid 4♥.

When your partner makes an overcall, and you hold:

0 - 5 HCP   Pass!

6 - 9 HCP   Support partner's suit if you have three cards; otherwise SHUT UP!

9-12 HCP   Jump Raise in partner's suit;
                Bid your own suit if it's a decent suit
                Bid No Trunps if you have a stopper in the opponent's suit.

13+ HCP     Raise to game in partner;s suit
                Bid a new suit if you can
                Jump Bid in No Trumps.