What is an active region in sun's photosphere? Why sometimes there is an active region but no sunspots? What is the theory behind an active region?  If we check data in solar monitors,  there are some cases where there is no sunspot but there is still an active region.
Why is that the case?
 A: Active regions are observed in EUV and x-rays while sunspots are observed in white light.  Thus, sunspots tend to be identified as photospheric phenomena and active regions coronal.  It can be the case that an active region is due to a sunspot, but it need not be.  See the link to the corona Wiki page and then look for the sub-header on Active Regions.
As an aside, the type of photons used to make an observation often correlate with altitude.  For instance, EUV and x-rays are typically found exclusively in the corona and white light down near the photosphere.  While both active regions and sunspots correspond to enhanced magnetic fields in spatially confined regions, they are seen at different altitudes.  Generally, a sunspot will have an active region overhead due to its enhanced magnetic fields, but an active region need not have a corresponding sunspot.

What is the theory behind an active region?

The plasma in the photosphere, chromosphere, and corona are tied to the magnetic field, something called the frozen-in condition.  Bulk fluid-like motions can cause magnetic fields to accumulate in spatially confined regions.  If the fields get strong enough, they will actually reduce the thermal pressure of the plasma by pushing out the hotter particles, resulting in a sunspot if the region gets large enough.
Active regions can be the precursors to sunspots or never form into sunspots and just embody locally enhanced magnetic fields.  Sunspots are regions where the magnetic pressure dominates over the particle pressure, pushing out hotter particles.  The total pressure stays constant, i.e., the magnetic pressure plus the particle thermal pressure.

Why is that the case?

Sunspots are an extreme, if you will.  They are the result of the magnetic forces dominating over the fluid forces.  Compared to active regions, they are rare.  Active regions are just regions of enhanced magnetic field identified as groups of loop structures seen in EUV and/or X-rays.  They form for the same reasons as sunspots but are just not as intense.
Update

Why can there be active regions but no sunspot?

First, what is a sunspot?  A sunspot is identified as a (comparatively) dark region on the optical surface (photosphere) of the Sun.  They appear dark because they are effectively a pressure balance structure, where the magnetic pressure is balancing the thermal pressure.  Here, it is easier to cool the plasma than reduce the density (i.e., density gradients are more unstable than temperature gradients), so the thermal pressure can decrease in response to an increased magnetic pressure (the density is a little lower too, but mostly the temperature is lower).  If this is strong enough, a large, dark region will appear on the surface.
So one might ask, why do they occur in the first place?  First, look at my derivation and explanation for the frozen-in or flux freezing phenomena in plasmas at https://physics.stackexchange.com/a/452325/59023.  I reference that answer as the photosphere is a region where it applies well.  That is, if the bulk of the plasma moves, the magnetic field will follow (very crude explanation but bear with me).
So one way a sunspot can form is due to a magnetic flux tube (i.e., just think of a cylindrical region of plasma with much higher magnetic fields) that is more bouyant than the surrounding plasma.  They rise to the surface and can develop a pressure-balance relationship with the surrounding plasma.  As discussed above, their higher magnetic field strengths require a lower thermal pressure to remain in pressure balance and since it easier to lower the temperature (again crude language) the main drop in thermal pressure is due to a lower temperature than a lower density.  Since the brightness of a black-body source can be related to the temperature, the cooler region is seen as darker.
Finally, we return to the original question.  Why can there be active regions with no sunspots?  This is relatively simple as an active region is just a weaker form of a sunspot in many ways.  The Sun is controlled by magnetic fields and plasma/fluid dynamics.  Some regions are dominated by plasma/fluid dynamics, others by the magnetic fields.  The photosphere is the region where these two start to "fight" for dominance, if you will.  The convection zone is where the plasma/fluid dynamics dominate.
Active regions are regions of enhanced magnetic field but that does not mean there will necessarily be a large region of cooler plasma due to the enhanced magnetic fields.  The fields here are stronger than the surrounding plasma, but not upwards of 1000 times stronger like that in a sunspot.
