My mother has a Dell and I am planning on getting a Mac because my PC is crap and got infected with a group of viruses. Apple seems the most secure and virus free computer. So I must know my mother’s computer has wireless internet, it would signal that my Mac to the Internet, too? Thank you! 
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Yes, it will be able to connect to the same wireless network.
But I would like to correct you on one thing, Apple does not carry the most secure and virus-free computers.
Apple is just as prone to security holes as a PC is, and they do have viruses and will increase in frequency once Apple gets a higher market share. Virus writers don’t want to write a virus that only operates on 10% of the the computers out there, and Apple computers only make up about 10%.
If they were the majority share, they would have just as many problems as Microsoft does.
If you have a wireless router in your house, your Mac would pick it up. If your mom can share a hard wired connection via her wireless card in the Dell, your Mac can pick it up. The only problem would be that your mom’s Dell would need to be on for you to get the signal.
The other person was wrong concerning Macs & viruses. There hasn’t been a MacOS X virus released in 4 years. There most likely is not a single Mac running Mac OS X with a computer virus right now in the world. The last virus died out after infecting 49 Macs. There are Mac trojans which are not viruses. You have to be duped into downloading and installing them with your admin password. They are in fake & illegal software downloads. The trojans are deigned to turn a Mac into a spambot. Roughly 25,000 Mac users were greedy and installed malware. There are over 10,0000,000 Windows computers with malware on them so they are used as spambots.