Dear Cornish Jack.
Wireless is not simple and never has been.
Consider that you can have a bunch of things affecting your radio waves, much like listening to a radio station and someone in a taxi parks up outside and keys up their CB radio. As a result of that, and the fact that your house is not likely to have been designed for wifi, means that you get NO guarantees of anything at all.
Now, with that said, wifi is actually one of the simpler forms of communication to understand:
You have a radio spectrum with 14 "channels" which sit next to each other in that spectrum. These channels are NOT identical the world over because, for example, the US Military demands use of some of the higher ones over in their country, whereas Japan etc has channel 14 and we in Europe don't.
You WILL get interference on your wifi connection if someone else in broadcastable distance is within 2 channels either way of your wifi in the spectrum (e.g. if you've chosen channel 6 then anyone broadcasting wifi on channels 4 to 8 will to a greater or lesser extent interfere with you).
(use a program called InSSIDer which is a free download to understand this more).
Wifi is a 2-way broadcast. Hence you have a wifi access point (typically your plethora of routers) AND you have your laptop or other wireless device. They don't just 'listen' but 'talk' as well, so don't forget that this may be some of the reason for issues in wifi as well (such as the recent Apple issues with firmware updates etc).
Wifi comes in several flavours, (802.11a, 802.11b, 802.11g, 802.11n, 802.11ac, etc) the most popular currently are 802.11g and 802.11n which give 54 and 150/300 mbit/s throughput. You need to match the output from your access point with the flavour of wifi your device wants to talk (or rather the other way round usually).
Now, you can broadcast your wifi out from your access point, but if you don't secure it with something then anyone and everyone can connect to it and use it (which is not generally a good idea), hence there is a logical layer of security overlaid on the top of wifi, and most of the time it'll be an incompatibility with negotiating security that'll give you problems.
There are 3 types of security in popular use:
WEP
WPA
WPA2
WEP is deprecated. Don't use it.
WPA is sort of deprecated too. If you have a choice between that and WPA2 then go WPA2, but if it's WPA or WEP then choose WPA (and if it doesn't include WPA2 then that might show that you need to look at a newer wireless device)
WPA2 is the security encryption of choice.
Along with that sort of security you get subsets of how you want it implemented, e.g. "pre-shared key, RADIUS server, certificate" etc. For 99.999% of all home users you want "pre-shared key" (which means that you program in a passphrase or key into the access point and you need to match that in the wireless device as well when you try and connect so that it negotiates a match and allows connection).
The management summary of the previous 3 paragraphs is to use WPA2/Pre-shared Key and choose "AES" instead of "TKIP" as another setting it'll ask you.
So, once you have all the above set in any access point and connect a device to it, it should work successfully.
Other things you may need to know about wireless are:
SSID is the broadcasted name of your wireless point - it can be set to anything you like. You may even get the opportunity to set several (but unless you know what you're doing, stick to the first one only).
Your passphrase (pre-shared key) can be anything you like, as long as it's over 8 characters. Ignore the techie guys here who'll argue the toss about security, as anything other than a dictionary-based word will require the would-be hacker to park a Cray outside your house for years before they crack it....and for what? a home broadband connection?
Incidentally, there IS a protocol called WPS which might be what you think you're after. However, it's a better marketing tool than a reality as it's not exactly ideal.
Wi-Fi Protected Setup - Wikipedia, the free encyclopedia
Now, armed with a lot more information than you possibly had previously about wifi, do you feel a little more like tackling the problem you have?
I'd hazard a guess that you weren't aware that wifi is a 2-way discussion, and that if you're replacing router after router you haven't looked at the other end which is possibly your wifi-enabled laptop or iPad etc.
Anyway, if you have issues, jot them down in as much detail as you can manage (detail helps us troubleshoot), and i'm sure any of us who know enough about wifi will give you a hand.
Cheers,
Mike.