Is possible some elements (like Cadmium or Bohr) absorb neutrons, without any nuclear reaction in their atomic nucleus? If the answer is YES, then why there isn't any nuclear reaction?

Isotopes have halo neutrons, that appear to pair up and have "extra-nuclear" orbits. Isotopes decay after a while.

Neutron absorber (http://en.wikipedia.org/wiki/Neutron_absorber)
There is a nuclear reaction, but the result is only to form a heavier, stable isotope of the same element.

Do they decay after some period? Is there big energy released?

No, it's the weak interaction. The decay half-life depends on the isotope's atomic number - how strongly coupled the halo neutrons are, presumably.

So, there is not any fission in the atoms of neutron abosorbers, right?

But why there is such an "explosive" reaction in the atoms of uranium-235 or elements used for fission?

Uranium is a much bigger nucleus, it decays differently to most isotopes (I think). Have a look at fission reactions, and isotope decay. Uranium fission involves alpha particles...?

Uranium fission involves slow neutrons.

Is half-life decay same with nuclear fission?

What is their period of decay, after receiving the neutron? Are control rods inside nuclear power plants often replaced because of this?

...here you go:
Many heavy elements, such as uranium, thorium, and plutonium, undergo both spontaneous fission, a form of radioactive decay and induced fission, a form of nuclear reaction.
Elemental isotopes that undergo induced fission when struck by a free neutron are called fissionable; isotopes that undergo fission when struck by a thermal, slow moving neutron are also called fissile. A few particularly fissile and readily obtainable isotopes (notably 235U and 239Pu) are called nuclear fuels because they can sustain a chain reaction and can be obtained in large enough quantities to be useful.
...
Not all fissionable isotopes can sustain a chain reaction. For example, 238U, the most abundant form of uranium, is fissionable but not fissile: it undergoes induced fission when impacted by an energetic neutron with over 1 MeV of kinetic energy. But too few of the neutrons produced by 238U fission are energetic enough to induce further fissions in 238U, so no chain reaction is possible with this isotope. Instead, bombarding 238U with slow neutrons causes it to absorb them (becoming 239U) and decay by beta emission to 239Np which then decays again by the same process to 239Pu; that process is used to manufacture 239Pu in breeder reactors, but does not contribute to a neutron chain reaction.

Fissionable, non-fissile isotopes can be used as fission energy source even without a chain reaction. Bombarding 238U with fast neutrons induces fissions, releasing energy as long as the external neutron source is present. That effect is used to augment the energy released by modern thermonuclear weapons, by jacketing the weapon with 238U to react with neutrons released by nuclear fusion at the center of the device.--wikipedia.org

But why they absorb neutrons? What is their half life period?

Are you asking the half-life of a neutron? It's about 15 min, but that depends how fast it's going.

Asking why neutron absorption is a little like asking why electron orbitals...? These things are "matter" particles that interact, is all. We explain interactions in terms of these things.

No, I am asking what is the half life period of the absorbers of neutron?

Try finding a table of isotopes somewhere, that's what an element that's absorbed a neutron or two is. The periodic table is a list of stable isotopes, or "things that represent the same thing". I think that's what "isotope" means, but don't hold me on that.

"Isotope is best used when referring to several different nuclides of the same element; nuclide is more generic and is used when referencing only one nucleus or several nuclei of different elements. For example, it is more correct to say that an element such as fluorine consists of one stable nuclide rather than that it has one stable isotope."

..is what the last edit in wikipedia reckons.

P.S. regarding your original question, "nuclear reaction" can mean quite different things apart from fission or fissile elements.

After all, electrons interact with nuclei, they're bound by the charge, of the nucleus.
But with just the "boundary" of a nuclide, and considering halo neutrons which are there somewhere "near" the denser conglomeration of protons and neutrons, or a stable nuclide core, because of neutron capture.

"One example of a halo nucleus is 11Li which has a half life of 8.6 mS. It decays into 11Be by the emission of an antineutrino and an electron. Its cross-section of 3.16 fm is close to that of 32S, a much heavier nucleus. Its radius is close to that of 208Pb. It contains a core of 3 protons and 6 neutrons, and a halo of two independent and loosely bound neutrons.

Nuclei which have a neutron halo include 11Be and 19C. A two-neutron halo is exhibited by 6He, 11Li, 17B, 19B and 22C. Two-neutron halo nuclei break into three fragments and are called Borromean because of this behavior. 8He and 14Be both exhibit a four-neutron halo." - from wikipedia, again.

And why in Uranium-238, there is fission with fast neutrons, and it absorbs slow neutrons, how come that?

Well, I think it has to do with potential states and barriers, or "allowed" states. For example, an alpha particle is tightly bound in a uranium nuclide, but somehow tunnels through the barrier, and alpha decay occurs - very slowly, but it's like there's a constant tension on this alpha particle, pulling on it elastically, and eventually - given sufficient time, the probability of it being on the other side of the barrier approaches unity.
Probability is like that.

When Uranium-238 is attacked with fast neutrons, do it absorb it ?

And can I ask, how can we keep the neutrons, which want to escape, in the chain reaction? In my book says that it can be done by neutron caputre or neutron diffusion. I don't understand what they mean. PLease help! Thanks.

You've asked about neutron diffusion elsewhere, haven't you?

Try to imagine what "diffusion" might mean in terms of neutrons: things diffuse by separating, rather than gathering or concentrating, so neutron diffusion is related to neutron density - like the density of gases that mix together. It's also a term used in scattering experiments, but scattering is just "interaction angle".
More scattered means - you guessed it - more diffused.

And how can we prevent losing neutrons with neutron capture?

Well, the capture rate would be related to the cross-section - if you surround a neutron source with a spherical shell of "neutron-capturing stuff", you should get a fair chunk of them. Or convert whatever it is into isotopes.

But I will capture them, lets say. How they will be released than, so they can be used in the chain reaction?

If you capture a neutron, it's captured, and depending on what type of nuclide has captured it, it will stay captured.

Neutrons aren't like things you can store and forward so much, more dampen down or squelch.

And why in Uranium-238, there is fission with fast neutrons, and it absorbs slow neutrons, how come that?

There is not fission with fast neutrons. Fast neutrons go through a nucleus too quickly to interact and be absorbed. No absorption: no unstable heavier nucleus and no resulting fission.

In a nuclear device, including power plants, the moderator, often heavy water, slows down the neutrons so they go slow enough to be in the nucleus for enough time to be absorbed. It is the half fast neutrons that are the active agents. The fast neutrons are ghosts that have little effect. The resulting heavier nucleus, if unstable, sooner or later fissions. The control rods are materials that absorb neutrons by forming heavier but relatively stable atoms. By absorbing neutrons and taking them off the playing field, fewer atoms absorb neutrons and become unstable.