The subject is new to me, let's say a company is offering to buy out a company with a definitive agreement to pay around $77 per share

China?s Qihoo 360 Moving Closer to $9 Billion Definitive Buyout Offer - WSJ

Does this mean if they do buy the company, will the price of the stock go to $77 and stay there for years, or can the price go higher, or can the price go lower?

Thanks

it can go higher
1° if other buyer seem to be interested and pay more
2° if short covering and people are absolutely paying more to buy shares

it can go lower, if the deal does not go through, or if the conditions change

the price will not necessary stay there
1° the stock can be delisted
2° there are techniques (go dark) to buy out minority share holder (reverse split/split)

hope this helps...