In the EU Parliament, there are three main conservative groupings:
1) The European People's Party. This is composed primarily of Christian Democrats. Parallels can be drawn with "Faith and Family" Democrats in the USA. They're moderately socially conservative, weak on civil liberties, opposed to drug reform, and will usually prefer shrinking welfare to expanding it, but they're also generally not advocates of a small state like most of the Republican Presidential Candidates were - they won't talk about abolishing healthcare programs very often, for example. Think: Angela Merkel
2) European Conservatives and Reformists. Less religious than the Christian Democrats (so often stronger on reproductive rights and LGBT+ rights), but broadly occupy a similar platform. Generally more Eurosceptic, but advocates of reform rather than abolishing the EU. Think: David Cameron
3) Europe of Freedom and Direct Democracy - a mixture of some far-right, highly conservative politicians, anti-politics parties like the Five-Star Movement, and hard-right Eurosceptics like UKIP. It's hard to pin a political position on these guys given that they're basically united by Euroscepticism and nothing else (the Five-Star movement tried to join two left-wing groups before settling for EFDD, which puts them rather at odds with Robert Iwaszkiewicz who is a monarchist and Holocaust denier), but generally they're both further to the right and more authoritarian than the other two groups... they'd just like domestic authoritarianism rather than dictatorship from Brussels.
There's also a far-right group but that doesn't have support from enough countries to be officially recognised (you need members from at least 7 or 8 member states). It would be a large group if it could get that one extra country though, particularly if it recruited UKIP. Prominent parties include the Dutch PVV led by Gert Wilders and the French National Front led by Marine Le Pen.