I'm staying at a relatives house for vacation but they do not have wifi. I have wifi at my house, If I take my router and plug in my equipment will I get internet access?
The answer is "It depends, but probably not".
If by WIFI you mean "Internet Access", and they don't have Internet themselves, the answer is "No". Briefly speaking, there is equipment on the other end which needs to be enabled on a per line basis - it will not be enabled on their line.
If they already have Internet, but no WIFI then * If they use the same ISP and type of connection, it is possible (note possible, not likely) that it will work. In this case, Internet will most likely be billed to your account. This makes all maner of assumptions which including stuff in the heart of the ISP network.
If your connection is provided over ethernet, ie your WAN port plugs into a regular ethernet jack, you can most likely take your WIFI device and plug it into their router. This is not guaranteed to work (due to IP address conflicts), but has a high probability of working or being able to be made to work by reconfiguring the IP addresses on your router.
If they do have Internet, but not WIFI, your best bet might be to actually buy an inexpensive Access Point or Ethernet router to provide the above functionality. You can get a new access point for less then US$30 from Amazon, and a second hand one even cheaper from Ebay.
I think some of the answers may have been a bit confusing with their use of the terms "router" and "modem", probably based on your question's unclear use of the word "router".
If your "router" was provided by your ISP, and it is the only box given to you by them, connected to a coaxial cable coming into your house, and you still have Wi-Fi, then it is a combination cable modem / wireless router (not a good idea from a security standpoint, so you should get your own separate cable modem and router anyway in that case).
In case you bought your own wireless router, and the network cable from its "WAN" port plugs into a cable modem provided by your ISP, then the wireless functionality comes from your router.
If you could clarify in your question what you mean by your relatives not having Wi-Fi, we could provide a more accurate answer.
Do you relatives have Internet at all? If they do, is it through a cable modem? In that case, you can simply take your wireless router and connect it to their cable modem. I recommend restarting the cable modem so it can detect the new router MAC address. If the relatives' ISP has locked the cable modem to the MAC address of whatever equipment (wired-only router, standalone computer) is connected to the modem, you can try accessing the GUI configuration page of your router, and set the MAC address of the router to the same address as the previously connected equipment.
Your relatives do not have Internet, but have the same cable company providing service to their house you have at your place? You could take your cable modem and your router, and establish service at their house.
However, there are a few issues with that, depending on the cable company. Some ISPs will not allow a cable modem that is still active on another account on their network start new service at a different location. This is in part to prevent someone using a stolen cable modem, racking up a bill for the original owner (in case he or she is on vacation and has not noticed the modem having been stolen).
Furthermore, there may be installation and setup fees, and you may also be charged an early termination fee if your home Internet plan was on an annual contract so you could benefit from a discount. Your ISP may allow you to transfer the physical location, but there are no guarantees, and then you will have to go through the same circus when you return home.
To clarify something mentioned in another answer: you will not get billed for connecting your router to your relatives' existing Internet access. You would only get billed if you connected your own cable modem to their Internet line and enabled service on it with your name and your cable modem.
If your friend does not have an existing connection to his/her house, then no you cannot just take your router and plug your equipment in. The modem/router is just an interface that enables your devices to utilise the internet connection coming to your house, it itself is not the device that brings internet to your house.
If however, your friend already has an internet connection running to his/her house, then in most cases you can take your modem over, hook it up and it should function as per normal.
Depending on the required range, you can also get equipment that will beam the connection from your house to another location.