Radical feminists (from the latin for "root") argue that because of biological differences (women are vulnerable during pregnancy and in the early years of childrearing, and are physically smaller and weaker than men because of the male testosterone surge in puberty), men have been able to push women into second place basically since... the beginning of the human race (there is argument about whether the oppression started with agriculture when passing your land onto your sons became important hence men wanted to control women's fertility, or whether hunter gatherer societies also have this sort of hierarchy - but it doesn't matter for the purposes of this argument). Radical feminists argue that you can't fix inequalities between men and women by tinkering round the edges, you need to dismantle the whole system
Liberal feminists (following on the liberal tradition in politics as a whole) believe in individual liberties up to the point at which you impinge on someone else's liberties - the "your right to swing your arm around ends where my nose starts" theory. Typically they tend to think that what we should be doing is to use existing political structures to make incremental changes to make men and women's life chances more equal. (One of the best explanations of political liberalism takes the form of a thought experiment: suppose you are trying to design society from scratch. You know that some people are luckier than others - they're born physically stronger, or brighter, or with more confident personalities... How do you organise society so that the less lucky people get a reasonably fair deal, while allowing the people who work hard to be rewarded for their hard work, and all the while keeping the extent to which the state is allowed to interfere in people's private lives to a minimum? It's fairly typical of liberalism that most liberals believe it's about balancing opposing needs of different groups within society, while giving everyone the best chance of getting ahead that you can, and keeping the "hand of government" as light as you can.)
Take for e.g. women struggling to get on in the workplace. A liberal feminist might campaign for workplace creches. A radical feminist might ask why we have a culture of presenteeism at work, why we think careers in the public sphere are more valuable than looking after children or having caring responsibilities, why society is set up to give the big financial rewards to work outside the home, but not redistribute wealth to cover those in caring activities...
I tend to find radical feminist critiques of what is wrong with the way we do things at present very convincing, but ultimately (because most of us in the West live in capitalist liberal democracies) that liberal feminism has more of a chance of actually changing things.
As a separate issue there's the question of what you mean by the word "gender."
Almost all radical feminists will agree that the primary meaning of gender is "social role or behaviour which society considers is appropriate to your sex." Additionally, they will argue that gender on this meaning is what's known as a social construct - i.e. it's not part of the natural world, but grows out of how societies organise themselves. For example, in the UK football is predominantly a game played by boys; in the US it's played by girls (hence a recent TV programme on which a father of a "transkid" said one of the early signs his child - born with male genitalia - was really a girl was that "she" liked soccer). And further more, it's a social construct which conveniently always seems to allocate the hard but undervalued work (looking after children, looking after elderly relatives) to women and the very highly paid work to men.
Transactivists typically use "gender" to mean an internal feeling of being a woman or a man (or anything in between, or whatever... this is the point where I start to get very confused).
And of course there's an additional layer of confusion, because gender has also come to be used in place of the word "sex" to talk about actual biology, because people think sex is a bit embarrassing.
So now we have a single word - "gender" - which can mean three things depending on the context in which it's used - "what society thinks are the appropriate roles for someone of your biological sex," "An internal feeling of womanliness or manliness", or "The same as biological sex, just we're a bit too embarrassed to say the word sex." End result: total confusion and lots of scope for people to talk past one another.
However, "radical" is often used as an insult, to mean extremist (and people using it this way will also then move on to insults like "feminazi", or - as on a recent thread - comparing radical feminists to members of ISIS - which of course is nuts when you think about it: one group wants equal rights for women, the other two groups are going round killing, raping, committing genocide). Hence "TERF" was coined by people who want to discredit anyone who thinks "gender" is a social construct. It's basically an insult which says "shut up, bitch" and is used as a form of social exclusion - once someone is described as a TERF on twitter or tumblr or most other social media platforms, they are then completely ostracised (because these platforms typically indulge in a form of guilt by association: "talked to a TERF about a recipe for chocolate cake? Then you're a TERF too!")