Like many pure metals, silver mostly holds itself together with metallic bonds. A metallic bond is a weak attractive force between metal atoms. (The explanation for where this force comes from is complicated and not perfectly understood--sorry!--but it involves the electrons belonging to these atoms and how they move around the entire metal. If you want to learn more, try looking up terms like "sea of electrons" or "electron cloud.")

Metallic bonds tend to be weaker than covalent bonds. So when we say that silver is soft, we mean that it can be shaped with a hammer more easily than, say, diamond, which is made of covalent bonds. Materials scientists have many different ways to talk about how strong materials are, using numbers like the yield strength, the shear modulus and the elastic modulus (also known as Young's modulus). Each of these mechanical properties has a different and very specific meaning, and you would need to know several of them to get a complete picture of how "strong" a material is in general.

Just as an example, let's compare the shear modulus for silver to the shear modulus for some other materials. Silver has a shear modulus of 30 GPa--fairly soft. But the shear modulus of steel is about 80 GPa--much harder--and for diamond the shear modulus is 478 GPa. (On the other hand, rubber is much softer--less than 0.001 GPa.)