Learn Spanish (Latin America): El testigo nota el tatuaje que tiene el hombre. - The witness notices the tattoo that the man has.

No, in Spanish you can never skip the relative pronoun que. In English, you can say 'the tattoo the man has', but in Spanish, you must always include the word for 'that': el tatuaje que tiene el hombre.

Why is the word order tiene el hombre instead of el hombre tiene?

After the relative pronoun que, it is very common in Spanish to swap the subject and the verb if the subject is a noun (like el hombre). Saying el tatuaje que tiene el hombre (literally 'the tattoo that has the man') sounds much more natural to native speakers than que el hombre tiene.

How do I know tatuaje is masculine?

Almost all Spanish words that end in -aje are masculine. This is a very helpful rule to remember for words like el tatuaje (the tattoo), el viaje (the trip), and el mensaje (the message).

If the witness were a woman, would she be la testiga?

No, the noun testigo is an exception and always ends in -o. If the witness is female, you only change the article: la testigo. Because we see el testigo here, we know this specific witness is male.