Great question. It comes down to structures depending on their function. In other words, things like size, shape, texture, and other things depend on what something does. Both plant cells and animal cells have a membrane around the outside. It’s flexible like a water balloon, but unlike the balloon, it lets certain things in and out all the time.

The plant cell has another layer, the cell wall. It’s made of the stuff that makes celery so hard to chew up. It’s like a stiff box.

Plant cell walls are what gives plants their structure so that they can stand upright. The walls also protect them from invaders. You know how a plant wilts when it needs water? The cell walls are still there, but they’re saggy. When the plant starts to take up water from its roots, tubes take the water all around the plant. Water goes into the cells. If the walls weren’t there, water would keep going in until the cells burst, like an over-filled water balloon. (I don’t know whether your class has studied osmosis yet, that’s why the water enters the cell.)

In order for things to move between the plant cells, there have to be holes in the walls. But the cell walls also have to be connected and packed tightly or the plant wouldn’t have any strength.

Some animal cells are tightly packed and stay in place, but some have space around them. It depends on what they do. Blood cells are flowing around in the liquid we call plasma. Moving oxygen around our body is one of the functions of red blood cells, so they are small, smooth, and can squeeze through tiny blood vessels. The walls of the blood vessels have to be pretty tight so that the liquid doesn’t leak out too much.

What other animal cells might have to be tightly packed or floating around?

An interesting question! That would be because plant cells have a layer around them called a "cell wall". The cell wall is a layer made of cellulose and gives the cell structure, protection, and filtration. It allows plant cells to pack closely together. Animal cells do not have a cell wall. At best, animal cells keep together through these connections called "cell junctions". Cell junctions provide some contact and adhesion between cells, but they are not as strong as a cell wall, so it is easy for them to separate and float around.

There are four types of tissues: connective, epithelial, muscle, and nervous tissues.

Blood and lymph, surprisingly, are fluid connective tissues. The red blood cells and platelets are in the blood while the white blood cells are in both the blood and the lymph. Those cells are in a liquid medium and they indeed float around. Since white blood cells are responsible for our immune system, some white blood cells such as the macrophage can active move toward the foreign particles and engulf them.

Beside those two types of tissues, all other animal tissues are connected together, sometimes very tightly even. For example, epidermis, the outermost layer of our skin, is composed of epithelial tissues. It is so tight that our skin is waterproof!

Blood cells look like they're 'floating' around because they're in the plasma of our blood. Blood plasma is the liquid that doesn't have cells. Red blood cells take up about half the volume [the space] in our blood, and there are some white blood cells too, but not nearly as many as red blood cells.

Animal cells in tissues are packed together. This link shows a nice picture of cells packed together and a nice picture of a slice cut through a tissue, which shows the different types of cells on the outside and the inside of the tissue, all packed close together.