Learn Swedish: Jag slog knäet mot bordet. - I hit my knee against the table.

In Swedish, you usually use the definite article (like knäet, the knee) rather than a possessive pronoun (mitt, my) when referring to parts of your own body, as long as it is clear from the context whose body part it is.

I have also seen the word for the knee spelled as knät. Is there a difference between knäet and knät?

No, they mean exactly the same thing! Ett knä is an ett-word ending in a vowel. Its definite form can be written either as knäet (the more traditional spelling) or knät (which is closer to how it is actually pronounced). Both are completely correct and widely used.

Is slog the past tense of slå?

Yes. Slå (to hit or strike) is an irregular verb. The core forms you need to memorize are slå (infinitive), slår (present), slog (past), and slagit (supine).

Why use the preposition mot here? Does it not usually mean towards?

While mot often translates to towards when talking about direction, it is also the standard preposition for against when referring to physical impact or a collision. So mot bordet simply means against the table.