Dovish refers to the stance and sentiment a government monetary policy creator holds and expresses toward keeping interest rates low. Doves on the Federal Reserve board believe low-interest rates and accommodative monetary policy are important to keep the economy going and maintain low unemployment rates. If someone is dovish they are not as concerned about inflation increasing as they are about keeping an economy expanding. When any economist disregards the current danger of the effects of growing inflation on business, savers, or the economy they are considered dovish. Doves call for lowering interest rates or keeping them low, quantitative easing, and sometimes asset purchases to hold up markets.
Overtime economists can become concerned that excessive dovishness can create asset bubbles and cause runaway inflation. The opposite of being dovish is hawkish, as hawks want to tighten monetary policy and avoid bubbles and dangerous inflation.
The stock market prefers the Federal Reserve to remain dovish as low interest rates provide cheap money to borrow and invest in business, real estate, and equities driving up asset prices.