If you want a (free!) glass of tap water, ask for un vaso de agua del grifo. If you're eating at a typical Spanish restaurant, this glass of water won't come with ice. Tap water in Madrid is completely safe to drink.
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For tap water ask for agua de grifo or una jarra de agua for a pitcher, and if you are happy with a bottle just ask for agua.
Tipping in Spain works differently: The customer actually has the option to not leave any tip at all. However, restaurants expect a 10% tip on each bill—although it's not a formal rule, and that percentage is much lower than tipping practices in other countries.
In Spain, you can simply say la cuenta, por favor (the check, please) when you catch your server's attention. Another common phrase many locals use is ¿Nos cobras cuando puedas? (literally Can you charge us when you can?). Either one is a perfectly fine way to ask your server to bring the bill.
A question often asked by foreign visitors when they come to Spain is “Is it okay to drink the tap water here?” The simple answer is yes, you can! Spanish tap water is considered to be 99.5% safe to drink, although the taste may vary somewhat according to the region.