The primary meaning of виза is visa, the document without which you cannot enter a foreign country. The US and Russia have a tit-for-tat game going on that esentially means Russians can't come to the US without obtaining a visa (which is an onerous process), and Americans can't go to Russia without going through an annoying process as well. Here's a picture of my visa for the summer, with portions grayed out for obvious reasons. It occupies an entire page of my passport:

Back in the Soviet period a visa was a three-part form that was not attached to your passport. When you entered the country, they stamped the visa and removed on part. When you exited the country, they stamped and retained the rest of the visa so that once you were gone, there was no proof in your passport that you had ever been in the Soviet Union. I never quite figured out the reasons for that. Here's a photo of my visa from 1988:

Виза is a perfectly regular first declension noun:

Before you get a visa to Russia, you have to get an invitation from a family, hotel or business. That's a lengthy process as well. And of course there are fees involved. Once you get to Russia, you have to register your passport/visa with internal immigration, which produces another document and involves other fees. Then by law you are required to have your visa and passport and registration on you at all times, and the police can stop you at any time and demand to see those documents. And of course if you passport, visa or registration is stolen, then it's a major pain to get them replaced. And of course there are fees involved. It's entirely amazing how many problems can crop up with these things. For instance...

That may sound ridiculous, but it actually happened to one of my students in ’92. We were in Moscow at the time. Our visa support was in Leningrad. I ended up having to head back to Leningrad to get a new visa for him.

That also may sound ridiculous, but in fact that's what I had to pay for the visa for this summer.

That, too, may sound ridiculous, giving a complete stranger your passport, but in fact one of our students did precisely that in ’89. She got to know the American Embassy pretty well in the process of getting a new one.

That, too, may sound ridiculous, but all our students have to have HIV tests to get a visa. I think this is part of the tit-for-tat. Up until January of 2010 the US also required HIV tests for people getting visas to enter the country. Now that the requirement has been lifted, it will be interesting to see if Russia lifts it as well.