Back in the old days of Soviet rule, everyone needed one thing to survive: блат. This word, so satisfying to say — try it! — meant pull, an "in," connections. Блат got you tickets to a closed screening of a Hollywood film. Блат got you a tin of powdery but potent instant coffee from India. Блат got your kid a place in a preschool. But when the planned economy fell, it took блат with it. Now if you want something, you go buy it. You don't need блат with the sales clerk.
But if блат disappeared, связи (connections) are eternal. And luckily for all of us language-hoppers, they are pretty universal, too.
Связи — note the plural — are connections, contacts, people you know, relationships. Я хочу устроиться на работу в этой фирме. Нет у тебя там связей? (I'd like to get a job in that company. Do you have any contacts there?)
Sometimes the relationship connoted by связи is one of patronage. As usual, context is everything: Он поддерживает очень хорошие связи с местной администрацией (He's been cultivating an inside track in the local administration).
Связь (singular) is some kind of connection. It might be invisible: На сцене она умела сразу установить связь с аудиторией (On stage she could immediately establish a bond with the audience). It might be a bond of friendship or professional interests between individuals, institutions or countries: Количество внутренних и внешних — межрегиональных и международных — связей нашей компании постоянно возрастает (The number of our firm's domestic and foreign — inter-regional and international — relationships is constantly growing).
Or it might be something more intimate. Связь, sometimes modified with the word любовная (amorous), is any kind of extramarital relationship. Её муж, не зная о прежней нашей связи, старается познакомить нас (Her husband, who doesn't know that we had an affair, is trying to introduce us to each other). When связь is modified by the adjective половая (sexual), we are talking about a physical relationship without much emotional attachment. And случайная связь (literally an incidental contact) is what English speakers call a one-night stand.
This should not be confused with обратная связь (literally a reverse connection), which is feedback. Обратная связь с клиентами жизненно важна для сохранения конкурентных преимуществ нашей фирмы (Client feedback is essential if you want to keep your company's competitive edge).
In the category of invisible connections, связь can also refer to the internal logic of speech or writing. Его воспоминания про войну проносились друг за другом без связи (His unconnected reminiscences about the war came tumbling out one after another). Sometimes this is expressed with an adverb: Он был в состоянии какого-то наркотического опьянения и совершенно несвязно что-то говорил (He was under the influence of some kind of drug and spoke incoherently).
Связь can also refer to forms and institutions of communications, like the railway, the post office or an Internet hook-up. Sometimes you need to clarify: На даче связь есть? (Do you have communications at the dacha?) could refer to roads, telephone lines or cell coverage, cable internet or even commuter trains. Да, беспроводная связь есть, но слабый сигнал (Yes, there's wireless internet, but the signal is weak).
Russians enjoy the joke-producing ambiguity of связь, especially when combined with another polysemous word like брак (marriage or defect). What's связь без брака? At the water cooler in a telecoms company, it's quality communications. In the locker room, it's an adulterous affair.
As usual, context is everything.