Here are the technical details:
The “soft sign,” Ь, indicates that you should palatalize the preceding consonant. That is, you touch your tongue to the hard palate (the roof your mouth near your teeth). This adds a sort of “y” sound to the consonant.
The “hard sign,” Ъ, used to be used primarily to indicate that you should not palatalize the consonant. Nowadays, it has the main purpose of forcing a syllable break.
You can hear the difference between a soft consonant and a hard consonant, as in the American English “do” and the British English “dew.”
There are a few words where the presence or absence of a soft sign changes the meaning of a word. For example,
у́гол means "corner," but у́голь means "coal."
Again, don't worry about this - the context will make your usage clear. Given a choice between sentences that translate to:
The bank is around the corner.
The bank is around the coal.
a native Russian will figure out the correct meaning even if you pronounce it incorrectly.
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