Null-hypothesis testing and likelihood-ratio testing In this book
Wickens, T. D. (2002). Elementary signal detection theory. Oxford: Oxford University Press.
You can read:

and this confuses me as I thought that likelihood-ratio testing was a procedure used to test null-hypothesis. Is likelihood-ratio testing different from null-hypothesis testing and if so, do you know where I can find information about this type of symmetrical test (likelihood-ratio testing)?
 A: Perhaps two real-world examples will help!
For hypothesis testing, we are making asymmetric decisions, where the "cost" of a false positive is generally higher than the "cost" of a false negative.  Suppose that our data suggests that there is a 80% chance that a drug is effective and only a 20% chance that the results could have been random luck.  That might not be enough to take it, especially if there are severe side affects, it costs a fortune, etc...
Contrast this with a signal processing example - was the last bit we received a 0 or was it a 1.  You might not know the a priori probabilities so you only have the ratio of odds, not the true odds.  Also note that the "cost" to the system in making a 0 to 1 error is not significantly higher than making a 1 to 0 error or vice versa.  Therefore, even if the odds ratio is only 1.0001 in favor of a 1 having been transmitted, the receiver will pick a 1 and move along.
Of course this might be a little different if you had a priori probabilities or you were using soft-decision based error correctino.
A: To answer your question of where to find more information about likelihood inference methods, Dr. Jeffrey Blume has an excellent tutorial in Statistics in Medicine.
Blume, J. "TUTORIAL IN BIOSTATISTICS: Likelihood methods for measuring statistical evidence." Statistics in Medicine 2002; 21:2563–2599. DOI: 10.1002/sim.1216.
PS: Try Googling "Likelihood methods for measuring statistical evidence" to find it.
A: To your questions

I thought that likelihood-ratio testing was a procedure used to test
  null-hypothesis. Is likelihood-ratio testing different from null-hypothesis testing?

Yes, likelihood-ratio testing is one of many hypothesis testing methods. It is named so because of its log-likelihood ratio statistic.
Hypothesis testing is a generic term, and null hypothesis testing is one and popular subset. Null hypothesis testing is not required if in many testing methods, an example of which is the signal detection method mentioned.
