How can I set / change DNS using the command-prompt at windows 8

First, the network name is likely "Ethernet", not "Local Area Connection". To find out the name you can do this:

netsh interface show interface

Which will show the name under the "Interface Name" column (shown here in bold):

Admin State    State          Type             Interface Name
-------------------------------------------------------------------------
Enabled        Connected      Dedicated        Ethernet

Now you can change the primary dns (index=1), assuming that your interface is static (not using dhcp):

netsh interface ipv4 add dnsserver "Ethernet" address=192.168.x.x index=1

2018 Update - The command will work with either dnsserver (singular) or dnsservers (plural). The following example uses the latter and is valid as well:

netsh interface ipv4 add dnsservers "Ethernet" address=192.168.x.x index=1

To change DNS to automatic via command, you can run the following command:

netsh interface ip set dns "Local Area Connection" dhcp

Here is another way to change DNS by using WMIC (Windows Management Instrumentation Command-line).

The commands must be run as administrator to apply.

Clear DNS servers:

wmic nicconfig where (IPEnabled=TRUE) call SetDNSServerSearchOrder ()

Set 1 DNS server:

wmic nicconfig where (IPEnabled=TRUE) call SetDNSServerSearchOrder ("8.8.8.8")

Set 2 DNS servers:

wmic nicconfig where (IPEnabled=TRUE) call SetDNSServerSearchOrder ("8.8.8.8", "8.8.4.4")

Set 2 DNS servers on a particular network adapter:

wmic nicconfig where "(IPEnabled=TRUE) and (Description = 'Local Area Connection')" call SetDNSServerSearchOrder ("8.8.8.8", "8.8.4.4")

Another example for setting the domain search list:

wmic nicconfig call SetDNSSuffixSearchOrder ("domain.tld")

There are little difference in command of adding AND changing DNS-IPs:

To Add:

Syntax:
   netsh interface ipv4 add dnsserver "Network Interface Name" dns.server.ip index=1(for primary)2(for secondary)
Eg:
   netsh interface ipv4 add dnsserver "Ethernet" 8.8.8.8 index=1

• Here, to know "Network Interface Name", type command netsh interface show interface
• 8.8.8.8 is Google's recursive DNS server, use it, if your's not working

To Set/Change: (as OP asked this)

Syntax:
   netsh interface ipv4 set dnsservers "Network Interface Name" static dns.server.ip primary
Eg:
   netsh interface ipv4 set dnsservers "Wi-Fi" static 8.8.4.4 primary
   netsh interface ipv4 set dnsservers "Wi-Fi" dhcp

• Last parameter can be none:disable DNS, both:set for primary and secondary DNS both, primary: for primary DNS only. You can notice here we are not using index-parameter as we did in adding DNS.

• In the place of static you can type dhcp to make DNS setting automatic, but further parameter will not be required.

Note: Tested in windows 8,8.1 & 10.