How should I interpret information about bond spreads that narrowed or widened the most?

Bond spreads can mean many different things and is contextual. I will give you a non-exhaustive list of some of the things it can mean within some context, and the most common usage I have come across as a fixed income trader at an investment bank. Note that a comment mentioned yield curve spread, but I have never heard a yield curve referred to as narrowing or tightening or widening, curve spreads tend to steepen or flatten (or get more or less inverted), so I will ignore this. Also The comment about bid/offer spread is valid but probably needs little explanation. Otherwise..

Government bonds: typically there are two spreads of note, versus swaps, also called an asset swap spread or relative to some more noteworthy benchmark, e.g. Italian bonds spread over German bonds, for example, called an inter-country spread. But again widening or narrowing is contextual.. Italian bonds widening versus Germany means they become cheaper, whereas an asset swap widening means the bonds get more expensive relative to swaps.

Credit Bonds: typically the spread is the credit spread and it is effectively an asset swap versus swaps. Credit bonds usually have higher yields than swaps and so widening means that the bond gets cheaper relative to swaps.

Inflation Bonds: There is a term called the breakeven spread which refers to the yield on nominal bonds minus the yield on equivalent maturity inflation bonds. It is generically the expected level of inflation until that maturity, called breakeven since if that is in fact the realised level of inflation the two bonds will have been worth the same, held to maturity. Widening corresponds to increased levels of expected of inflation.

In the general case, if I wasn't sure, I would guess that "widening" corresponded to a cheapening in that specific bond, whereas "tightening" more likely than not means getting more expensive relative to whatever it is being spread against.

There are several usages of the phrase bond spread.

In the simplest sense, a bond's spread is the bid/ask spread of a single bond.

More commonly, bond spread refers to the interest rate differential between two bonds which allows one to compare the value of one bond to another. The higher the spread, the higher the risk of one of the issues.

A common usage of bond spread is the comparison of the yield of a particular government bill or bond versus bonds issued by corporations or municipalities. Even the spread between different countries is available. This data is provided on a daily basis and is also available historically.

For example, from the WSJ, here's a comparison of global bonds:

Yields and spreads over or under U.S. Treasurys on benchmark two, five and 10-year government bonds in selected other countries; arrows indicate whether the yield rose or fell in the latest session: (see link for data)

Bond spreads are also used as economic indicators. Widening spreads indicate a slowing economy and vice versa.