A currency pair like USD/INR is read as "how much of the second currency (INR) it takes to buy one unit of the first (USD)." USD is the base currency, INR is the quote currency, and the rate — say, 83.5 — means 1 USD buys 83.5 INR. This ordering isn't arbitrary; USD/INR and INR/USD are reciprocals of each other, not the same number written backwards.
"The pair moved up" means the base currency strengthened against the quote currency. If USD/INR rises from 83.5 to 84.2, the dollar got stronger, or equivalently, the rupee got weaker — it now takes more rupees to buy the same dollar. This trips people up constantly because "up" sounds like it should be good news for whichever currency you're rooting for, but it's specifically good news for the base currency and bad news for the quote currency.
Currency pairs get informally sorted into three tiers based on how much they trade. Majors involve the US dollar paired with a handful of other major economies — EUR/USD, USD/JPY, GBP/USD, USD/CHF. These have the tightest spreads and the most liquidity because so much global trade and investment flows through them. Minors, or crosses, pair two major currencies without the dollar in between, like EUR/GBP. Exotics pair a major currency with a currency from a smaller or developing economy — USD/THB, EUR/ZAR — and these tend to have wider spreads and can move more sharply on local news that wouldn't register globally.
What actually moves a pair, in rough order of how often it matters: interest rate decisions and expectations from the two countries' central banks, since capital tends to flow toward currencies offering a better return; inflation data, because it shapes what a central bank is likely to do next; trade balances and major exports (a currency tied to oil exports, for example, tends to track oil prices); and political events that change how safe or risky a country's assets look to hold.
If you're comparing two currencies you don't check often, look at both directions before drawing a conclusion. A currency pair page here shows the rate one way; the reciprocal page shows it the other. They describe the same relationship, but the number itself will look completely different depending on which currency is on top — 1 USD = 83.5 INR versus 1 INR = 0.012 USD are the identical fact stated two ways.