If the general market index goes down, is there a concept of an ETF that will go up on those scenarios?
Yes. What you're asking about are called inverse ETFs. Sometimes these inverse ETFs are also leveraged to provide 2x, 3x, etc. of the downside. But, beware:
An inverse ETF as any leveraged ETF needs to buy when the market rises and sell when it falls in order to maintain a fixed leverage ratio. This results in a volatility loss proportional to the market variance. Compared to a short position with identical initial exposure, the inverse ETF will therefore usually deliver inferior returns. [...] (Source: Wikipedia. Emphasis mine.)
Inverse ETFs aren't hard to find – they trade on the same stock exchanges regular ETFs trade on. You should have access through your broker if you can already buy ETFs. If you're interested in a list of specific inverse ETFs, here's one. Be careful with these.
Some related Q&A here:
You simply need to look for ETF names with words like short or bear / bearish those are the ones that go up when the market goes down.
Their counterparts are the ones that gain twice the market when it increases, but they also loss twice when it decreases. In these ETFs you usually find the word leverage / leveraged, because they basically borrow money (leverage) to invest more in the market.