There are several related concepts here. I'll define them first for clarity:

• Level: A line or plane is aligned horizontally, perpendicular to the axis of gravity
• Plumb: A line or plane is aligned vertically, parallel to the axis of gravity
• Square: Two lines or planes meet at a right (90 degree) angle
• True: Everything that should be level, plumb, or square actually is, in all relevant dimensions.

Plumb is probably the easiest of the four to check. You tie a weight at the end of a piece of string and dangle it from somewhere near the top, and the weight pulls the string into a straight line directly downwards. The word "plumb" comes from the Latin word "plumbum", meaning the metal lead, which then as now is commonly used for weights and sinkers because it's cheap, dense, and easy to work. The tools and techniques here are ancient: Egyptians were using recognizable plumb bobs as early as the III dynasty, about 4600 years ago. The ancient Egyptians generally used stone for their weights here.

Square is also relatively easy. The basic tool is a Try Square: an object with two arms meeting at a right angle. These can be constructed a few different ways. The most common modern way to do so exactly (apart from copying another known-good right angle) is pythagorean triples: right triangles for which all three sides are integer-length, such as the 3-4-5 right triangle (the two legs meeting at a right angle are 3 and 4 units long, and the hypotenuse is 5 units long). At least a few specific examples of right triangles were known in ancient times (e.g. the Plimpton 322 tablet, from Babylon c. 1800 BC), and Euclid's works (published c. 300 BC) contain a formula for generating pythagorean triples. I don't know if the ancient Egyptians used this particular technique, but they are believed to have used Try Squares also around the time of the III dynasty.

Once you have a way of measuring both Plumb and Square, you can use them together to check Level, since a level object and a plumb object will definitionally be square to one another if they meet up. The simplest way is to hang a plumb bob from below or from an edge of your level surface and use a square to construct level from your plumb line, or from your surface and then make sure the plumb line aligns with the vertical leg of the square.

One step better than this is to use a tool that's a combination of plumb bob and square, such as the A Square (modern English term) or Libella (Latin term). This is a an object similar to a try square, but with a third side joining the two legs an equal distance down each side, looking like a very wide capital A. The midpoint of the third side is marked, and a plumb bob hangs from the right angle vertex. When the plumb line hangs across the midpoint mark, that means 1) the two sides are both half-way between (45 degrees) vertical and horizontal, and 2) the cross-piece is level. Like Plumb Bobs and Try Squares, A Squares were also used by the ancient Egyptians.

The Romans also used a variation, called a chorobates, which had two plumb bobs at opposite ends of a long board, with alignment marks for each plumb line to indicate when the board was level at both ends. In addition, these sometimes also had a groove on top into which water could be poured: if the water fills the groove evenly rather than running out one side, then the board is level, at least along the axis of the groove. This was to allow the tool to be used if there was some obstacle (such as it being too windy to get a clean read with the plumb bobs, or not enough space for them to hang) to using the plumb lines.