Sometimes it's good to just set the record straight, and with that intention, let's take a look at zählen and zahlen. Short and sweet: zählen is "to count," and zahlen is "to pay."

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Sie zählen die Stimmen nach dem Ende der Wahl.

They count the votes after the end of the election.

Caption 4, Bundesrepublik Deutschland - Einbürgerungstest - Part 6

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Das macht zusammen vierzig Euro fünfzig. Zahlen Sie bar oder mit Karte?

All together that is forty euros and fifty cents. Will you be paying in cash or with a card?

Caption 7, Nicos Weg - Folge 16: Zahlen, bitte!

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Although you might be able to come up with a mnemonic device that you find helpful, simply remembering that Ich möchte zahlen means "I would like to pay" may be enough to help you sort it out. At the same time, you might still find that you get flustered mid-sentence trying to remember which verb is correct. It doesn't help that the noun die Zahlen means the "the numbers." It is the plural version of the noun die Zahl:

Der Adventskalender hat vierundzwanzig Türen, und auf jeder steht eine Zahl.

The Advent calendar has twenty-four doors, and each one has a number on it.

Caption 11, Eva erklärt - den Adventskalender

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In English, "to count" can also mean "to matter." Zählen is also used to mean "to count," "to count as" or "to number among." Take a look at these examples: 

Beim Blocken zählt jede Sekunde.

When blocking, every second counts.

Caption 37, Olympische Spiele - Beachvolleyball

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Sie zählt mit zu Deutschlands erfolgreichsten Sängerinnen.

She is counted among Germany's most successful singers.

Caption 2, Cassandra Steen - Geheimnis für positive Energie

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And one final note: you probably have wondered what the difference is between zahlen  and bezahlen. The truth is that they are often interchangeable, but there are some instances where zahlen is wrong. For example, when you speak about paying a person for a service, you would generally use bezahlen and not zahlen. 

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Further Learning
Look up the meanings of the related verbs anzahlen, zurückzahlen, and auszahlen, and search Yabla German to find them used in a real-world context. Some words related to zählen are die Volkszählung ("census") and der Stromzähler ("the electricity meter").

Unlike waitresses and waiters in restaurants in the United States, who often legally receive a wage well under standard minimum wage, but make up the difference through tips they receive from customers, waitstaff and bartenders in Germany usually receive at least standard minimum wage. In some cases, German waitstaff also earn a small percentage of their food and drink sales, regardless of the tips they receive from customers. 

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Perhaps it is for this reason that many German tourists, unaccustomed to tipping very much in restaurants and bars at home, often find it difficult to pay the standard 15 to 25 percent tip when visiting the United States. Germans often don't seem to understand that unlike in Germany, US waitstaff make their living primarily from tips, and merely rounding up the bill to the next dollar or fifty cents is not going to help them pay their rent.

The situation is so bad that in very expensive cities like New York, some waiters are reluctant to serve German tourists when they come into the restaurant, knowing from past experience that they may not earn any tip money from them. Since waiters are often taxed on the presumption of a 15 to 25 percent tip, they might even lose money if a table does not tip them. 

Let's turn the tables for a moment, though, with apologies for the pun. As an American in Germany, you will find most waitstaff extremely gracious when you give them a decent tip. You are not really expected to tip beyond a few euros, however, though I always tip 10% when the service is at least acceptable.

But what do you say when you wish to leave a tip? Best not to leave der Tipp, as that does not mean "tip" in the sense of tipping the waitstaff, but rather "tip" in the sense of a hint or advice, like a "hot tip for the racetrack." The correct German word is das Trinkgeld, literally "the drink money." There are several more subtle ways to tell waitstaff that the money you are giving them is for a tip than to use the word Trinkgeld, however.

Das macht zusammen vierzig Euro fünfzig. Zahlen Sie bar oder mit Karte? Äh, ich zahle bar. Hier sind fünfundvierzig Euro, das stimmt so. -Danke.

All together that is forty euros and fifty cents. Will you be paying in cash or with a card? Uh, I'll pay in cash. Here is forty-five euros, that's correct thus [idiom, keep the change]. -Thanks.

Captions 7-8, Nicos Weg - Folge 16: Zahlen, bitte!

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When Lisa says "das stimmt so" in the video above, she more or less means "keep the change," meaning the restaurant can keep as a tip the amount of money she gave them above the actual amount of the bill. However, if Lisa had handed them a 50 euro bill but only wanted to tip them 4 euros and 50 cents (for a total of 45 euros), then she could have stated the total amount she wanted them to keep and simply said "45 Euro, bitte."

So if you are paying at a cafe in Germany and the total for your coffee and cake is 9 euros and 20 cents, you could hand the waitstaff a 10 euro bill and say "das stimmt so," or you could hand them a 20 euro bill and say "10 Euro, bitte" to let them know that they can keep the amount over the bill up to 10 euros.

Additionally, you should be very careful with your use of the word danke when handing over cash: this will often be interpreted the same as das stimmt so. Don't be too surprised then if you thought you were merely being polite, but then the waiter doesn't return with your change!

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Further Learning
Read this German Wikipedia article on Servicewüste Deutschland, and see if you can understand it all without having to peek at the English version. Then go to Yabla German to search for "restaurant" to find some examples of German used in a restaurant context. For more restaurant vocabulary, see this article about eating at restaurants in Germany.