Republicanism pertains to the rule of law, as opposed to being ruled by a monarch or aristocratic elites who are "above" the law. Republicanism has a different meaning if one is referring to the party instead of a form of government, and to my knowledge there are no republican democracies these days, though many republics are representative democracies on paper - while in practice operating as aristocracies or oligarchies that to some extent are beholden to share power with and procure permission under the "social contract" to exercise legitimatized authority over folks.
Liberalism can refer to limited government based on a premise that the purpose of state is to protect the rights of citizens without taking unneeded steps to curtail the autonomy enjoyed by citizens as they go about their day-to-day lives. A "liberal democracy" tends to imply a republic or constitutional monarchy in which there is sufficient freedom to openly express oneself, assemble, associate, practice ones religion, and so forth without being persecuted as to foster what non-authoritarian capitalists consider a "free society." It is in contrast to an "illiberal democracy," where a government creates an outward facade of respecting human rights and representing the will of the people but is, in fact, characterized by a number of authoritarian tendencies.
Right-wingers often see liberal ideals in terms of "moral liberty," which is to say that people have the right to do whatever they want so long as it is moral, civilized, non-savage, etc. whereas centrists and people to their left tend to look at it more as a strategy of finding policies to implement which encourage a modernistic notion of "the good life" - passing regulations to protect people from harming one another and taking steps to produce an environment in which there is greater equality of opportunity and most, if not all individuals are free to exert considerable influence over their own courses through life. Or one may be talking about fiscal matters, in which case "liberal" is more or less synonymous with advocating policies friendly toward businesses, private property rights, and conducive to the voluntary exchange of resources between people.
Incidentally, some of the most modern interpretations of liberalism come striking close to principles of social democracy, insofar as the good, free, fulfilling life may demand the state - aside from upholding political and economic rights - do things like provide for the people a clean environment, sustainable economy, balanced budget, and the ability to practice their cultures unobstructed provided they do not call for behaviors that would inflict substantial harm upon others.
Mind you, that is not to be confused with liberalism as a political attitude, which stands in contrast to conservatism. A liberal in that sense is someone advocating patient, non-violent, gradual adoption of new ideas and reforms in government via existing social institutions.
Or something like that. I am a little fuzzy on this stuff so maybe someone else can explain it better.
Thank you, that was pretty helpful.
