What are the common names of this two types of plastic?
The most common types of plastics in use today are polyethylene (PE) and polystyrene (PS). Each of these comes in several variants.

It is difficult to tell just from a photograph exactly what plastics are used in the products shown. 

Also, various additives are often added to plastic to enhance certain properties such as high-impact, chemical resistance, recyclability and flexibility.

The type of plastic is sometimes marked on the item itself to help with recycling, typically in the form of an extruded imprint, or you could try to contact the manufacturer.

Both PE and PS are suitable for moulding and extrusion. You can also do some simple physical and/or chemical testing to determine what the plastics contain, although this will involve destroying at least a part of the item. Low density polyethylene (LDPE) typically has a lower melting point $(105-130^{o}C)$ than polystyrene $(200-240^{o}C)$.

LDPE is also softer and slightly rubbery whereas polystyrene is typically more rigid &amp; crackly when crushed. Without physically touching the items to get a proper feel for what they are made of, as a first guess, I would say that the colourful containers such as plastic bins and laundry baskets in your first photo look like some sort of polyethylene, whereas the clear plastic packaging in the second photo looks more like polystyrene.