Can someone explain this to me about the judiciary?

My understanding was that because of Parliamentary Sovereignty, the Supreme Court cannot strike down any laws, but today my teacher was talking about Blair's terrorist detention laws being "rejected" by the Supreme Court, but what can they do? Similarly, I seem to read the Factortame Case lead to the triumph of EU law over UK law, with a UK law at odds with EU law at the time, and the UK law was invalidated, or repealed, or what? This I just don't understand what happened and what happens with both courts. The HRA is also a little shaky, as it seems to be considered more valid than other laws? Does suggest that there are some laws that are considered more important then others, so there is a dualism in the constitution of constitutional laws and other laws? Might this be entrenchment? Any help on this would be appreciated on this little ramble!

The factortame case is just an example of UK law in volition with EU law. EU law's supremacy was first highlighted in the Treaty of Rome (if memory serves), so in effect you're right. Although parliament always remains sovereign (it's the basis of our entire political system) whether or not the UK is a member of the EU, after-all the UK would not be able to withdraw from it otherwise, also the Prime Minister is an MP so she is a member of both Parliament and Government etc. etc. The UK law (had to be repealed or would most likely face sanctions from the EU through a hearing by the European Court of Justice (ECJ) which had previous rulings on the case. The HRA however has nothing to do with the EU, the HRA just incorporated the European Convention of Human Rights which is made by the Council of Europe (not to be confused with the European Council or the Council of EU/Council of Ministers which are bodies of the EU) which doesn't really have any jurisdiction (arguably) in any nation bound to it (see the constant erosion of civil liberties in Russia). I don't know if this fully answered your question or not. But I hope this sort of helped.

Edit: Forgot to reply to the Supreme Court question
The Supreme Court effectively has no power in the British political system due to Parliamentary sovereignty as Parliament has the whole power of the political process. It rules, but parliament can ignore the rules (probably at its own peril though). It can get rid of local councils, merge local councils, get rid of regional assemblies and parliaments like those in Wales, Scotland and Northern Ireland and can get rid of the Supreme Court. It's important to note Parliament made all these institutions too (E.g. Constitutional Reform Act (2005) established the Supreme Court in 2009 and the Scotland Act (1998) created the Scottish Parliament). But as the Supreme Court was only established in 2009, and Blair had resigned by then and Brown was PM he may be talking about the Law Lords possibly. I can't say terrorism prevention is one of my strong suits.

Good point, it must have been the Law Lords then, but I assume it's the same if not similar to the Supreme Court. So then when the Supreme Court rules against a law for whatever reason, Parliament could in theory ignore them totally and do what they want? Over such a ruling, however, they tend to willingly but grudgingly repeal the law in question? If so, then how does the relationship work with the executive? Could the case over notifying the triggering of Article 50 (was it through the Supreme Court, I don't remember?) have been ignored or was it compulsory on the government because they are the executive while the legislature, Parliament, has the unlimited sovereignty? If it in the reverse was ruled that Parliament could not trigger Article 50 and instead the PM must do it, could Parliament have ignored the ruling and passed the bill of notification anyway?

My question about the HRA was its relation to the Supreme Court/Law Lords? It seems to me that if a law contradicts the HRA then that law is considered invalid by the Supreme Court/Law Lords, which I don't understand. It just confuses me substantially with what it can do and its power. I thought that in the event of a contradiction between one law and a newer law, the newer law supersedes the contradictions of the older one and becomes the new law, so surely any law passed after 1998 that contradicts the HRA supersedes the HRA?

Good point, it must have been the Law Lords then, but I assume it's the same if not similar to the Supreme Court. So then when the Supreme Court rules against a law for whatever reason, Parliament could in theory ignore them totally and do what they want? Over such a ruling, however, they tend to willingly but grudgingly repeal the law in question? If so, then how does the relationship work with the executive? Could the case over notifying the triggering of Article 50 (was it through the Supreme Court, I don't remember?) have been ignored or was it compulsory on the government because they are the executive while the legislature, Parliament, has the unlimited sovereignty? If it in the reverse was ruled that Parliament could not trigger Article 50 and instead the PM must do it, could Parliament have ignored the ruling and passed the bill of notification anyway?