I want to know the difference between Forward(Conditional) and Forward(LR) in variable selection method in logistic regression.

Here is quick explanation from IBM :

https://www.ibm.com/support/knowledgecenter/ko/SSLVMB_23.0.0/spss/regression/logistic_regression_methods.html

• Forward Selection (Conditional). Stepwise selection method with entry testing based on the significance of the score statistic, and removal testing based on the probability of a likelihood-ratio statistic based on conditional parameter estimates.
• Forward Selection (Likelihood Ratio). Stepwise selection method with entry testing based on the significance of the score statistic, and removal testing based on the probability of a likelihood-ratio statistic based on the maximum partial likelihood estimates.

I cannot find any relevant info about "probability of a likelihood-ratio statistic based on conditional parameter estimates" and "probability of a likelihood-ratio statistic based on the maximum partial likelihood estimates".

Can anyone explain or at least direct me to the textbook/paper that those procedures were based on?

Take a look at the LOGISTIC REGRESSION section of the Algorithms manual, which you can find here

ftp://public.dhe.ibm.com/software/analytics/spss/documentation/statistics/26.0/en/client/Manuals/IBM_SPSS_Statistics_Algorithms.pdf