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The circles in the image don't have the same center, so they are not concentric. Is there a word to describe circles that overlap (completely or not) so that enclosed circles are smaller than the outer circles?

In your particular picture, those circles are tangent circles (i.e., they are all tangent to each other). More specifically, they are internally tangent circles. According to Wikipedia, because they all intersect at a single point, these are also known as "kissing circles" (informally, I presume).

Of course, there is more than one way to draw circles that are "nested but not concentric":

I would call the circles in A nested circles; and the circles in B and C internally tangent circles. Because the circles in B happen to be tangent at a single point, they could also be called kissing circles. I would call the circles in D nested circles, but, because not all of them are tangent, I would not call them tangent circles.

If you want more formal terminology, perhaps math.se would be the place to ask.

I'm going a bit more into maths and geometry but the circle family in the image is called a parabolic pencil. These kind of circle sets are called pencil of circles. It is also a part of Apollonian circles.

A parabolic pencil (as a limiting case) is defined where two generating circles are tangent to each other at a single point . It consists of a family of real circles, all tangent to each other at a single common point. The degenerate circle with radius zero at that point also belongs to the pencil.

Also, they might be coaxal (or coaxial) circles.

Except for the two special cases of a pencil of concentric circles and a pencil of coincident lines, any two circles within a pencil have the same radical axis, and all circles in the pencil have collinear centers. Any three or more circles from the same family are called coaxal circles or coaxial circles.

Further information:

A pencil of circles is another name for specific families of circles who all share certain characteristics within the family.

The third kind of pencil is the parabolic pencil. This is the family of circles which all have one common point, and thus are all tangent to each other, either internally or externally. Also, the orthogonal set of circles to a parabolic pencil is another parabolic pencil.

_{Source: http://www.math.washington.edu/~king/coursedir/m445w04/as/ga2/gp2/groupassign2.html}

Coaxal circles are circles whose centers are collinear and that share a common radical line. The collection of all coaxal circles is called a pencil of coaxal circles (Coxeter and Greitzer 1967, p. 35). It is possible to combine the two types of coaxal systems illustrated above such that the sets are orthogonal.

A representation of Apollonian circles:

Eccentric: Not placed centrally or not having its axis or other part placed centrally.

I just found out that in topology, this is known as a Hawaiian earring

If you are mostly interested in a mathematical jargon term that generalizes to other instances, I think the best term would be descending. Whenever you see 'descending' in mathematics, it usually means something is getting smaller or is being nested in something else. Some common turns of phrase include descending sequence or descending sequence of sets. With your picture, I'd be able to say descending set of circles to some colleagues and be perfectly understood (this also applies to all of the pictures in J.R.'s answer).

In topology (a branch of mathematics where we study such shapes), we call it the Hawaiian earring.

Source: Mathematics student, have taken courses on topology.

Without the requirement that the circles be of decreasing size as shown, circles which overlap are simply intersecting.