There are situations when وَ‎ expresses “when”, “with”, or “while”. Such constructions with وَ are often difficult to translate as we need different words in English.

This happens in two situations:

The وَ is a واو الْحال (waw al-hal) and part of a circumstantial description.
The وَ is part of a مَفْعُول مَعَهُ (mafoul mahu), i.e., an object of accompaniment.

The Waw is a واو الحال

سافَرَ زَيْدٌ وَهُوَ طالِبٌ فِي كُلِيّةِ الطِّبِّ إِلَى مِصْرَ

Zayd traveled to Egypt when he was a student in the Faculty of medicine.

Note: This sentence does not mean: Zayd, who is a student in the faculty of medicine, traveled to Egypt.

If you want to express that, you should use dashes (…- سافَرَ زَيْدٌ – وَهُوَ طالِبٌ).

Some more examples.

.جاءَ الطِّفْلُ وَهُوَ يَبْكِي

The child came crying.

.تَخَرَّجْتُ فِي الْجامِعةِ وَأنا فِي الثّانِيةِ وَالْعِشْرِينَ مِنْ عُمْرِي

I graduated from university when I was 22 years old.

The Waw is part of the مفعول معه

.سِرْتُ وَالنَّهْرَ

I walked along the river.

اِسْتَيْقَظْتُ وَأذانَ الْفَجْرِ

I woke up with (at the same time of) the call to prayer.

A Wāw of accompaniment (واو المَعِيّة) can't share the action with what's before it. Watch out: Don't mix it up with the Waw al-Hal!

We know that in our example, the وَ introduces an object which the subject (in our example: I) is doing actions next to. Hence, we call it an object in connection with which something is done (مَفْعُول مَعَهُ).

Nevertheless, the most common application of وَ is to convey the meaning of and (واو الْعطْف). Even if you don't understand the words in a sentence, you can get closer to the meaning by looking at the cases.

1	مَفْعُول مَعَهُ	I walked with Zayd.	سِرْتُ وَزَيْدًا
2	مَعْطُوف	I and Zayd walked.	سِرْتُ وَزَيْدٌ

Remark: In the book Arabic for Nerds 2, I deal with stuff like that extensively.