I was reading this article about Marine Le Pen stating that
[Ms. Le Pen] vows to... strengthen ties Vladmir Putin's Russia.
What is so significant about strengthening ties with Russia, and why Vladmir Putin's Russia?
I also recall news making the headlines on Trump desiring a pro Russian American relationship. Is this something similar?
There would be several benefits :
the exports towards Russia suffered badly because of the sanctions, so lifting the sanction bilateraly would help the French economy, and especially the agricultural sector. See this pdf (please note that many people in the agricultural sector in France vote for Le Pen, so announcing she will lift the sanctions will gain her more votes).
Europe in general needs an access to Russian resources, especially gas. It is really damaging for Europe to let the cheap gas go instead to China, and France has nothing to gain from helping the USA and Germany in their political war against Russia. See this : with the EU sanctions, we literally pushed Russia toward China, and China benefits greatly while our economies suffer.
the French defense industry took a big hit as well as France had sold two Mistrals to Russia before the sanctions, and got pushed into cancelling the deal by the USA. France lost billions of euros because of that alone, see this (Russia may stop buying more French military tech, and they already started producing their own Mistral copies).
since De Gaulle, and until Sarkozy, there was the doctrine that France should align neither with the USA nor with Russia. The idea is that France has nothing to gain from siding with one against the other, and that it would be more interesting economically as well as politically to have a place on our own in the international stage. She could thus gain votes from people who cannot stand the current situation where France is follows whatever the USA tells it to do.
Russia can help resolve the conflicts in Syria, and thus stop the influx of refugees in Europe. In fact without Russia Syria would be completely in ruin by now.
On the other hand, I have trouble finding what would be the downside of having a good relationship with Russia (except angering the USA).
Politicians do not always present opinions which are their own. They try to present opinions and policies which may attract votes, and sometimes use controversial stances on certain issues as a way of gaining visibility (if you want votes, the voters have to know you exist and are different from your opponents). So:
It's not about what Le Pen thinks, it's about her presenting policies which may attract votes. Maybe she really believes in the need to improve Franco-Russian relations, but that is not important. Part of the French public wants to abandon the economic sanctions against Russia, since the Russian counter-sanctions are hurting some sectors of French economy. Another part of the French public wants to decrease the risk of possible military confrontation with Russia. Yet another part of the French public doesn't like US interventionism (blames it for the current European refugee crisis) and, in light of the evolution of the Syrian conflict, sees Russia as a natural counter to it. Le Pen decided to offer these voters what they want (apart from other things like the destruction of EU and islamophobia).
Yes, Trump's offer was in some ways similar, though he didn't talk much about the sanctions - US-Russian trade is miniscule compared to EU-Russian trade - and concentrated on what the US public fears post 9/11 - international terrorism. He argued that by cooperating with Russia, US would be able to more effectively fight ISIS and other terrorist groups, which would lead to decreased threat of terrorist attacks on US soil. The possibility of military confrontation with Russia is also a significant issue for US voters, if they don't know anything else about Russia, they know it's got nukes. Also, the currently trending russophobia is seen by some as dangerous to US democracy.
Why Vladimir Putin's Russiais an easy one... currently there is no other Russia to chose from. "Vladimir Putin's Russia" is just a figure of speech that reminds that nowadays relations with Russia means relations with Vladimir Putin and his government(in the same way that "relations with Germany" means relations with Angela Merkel and her government).