Air Cleaner Filters
You can get portable 'air cleaner' units that use filters to take particles out of the air. While these may frequently be less expensive than other kinds of air filtration systems, they also have significantly lower performance. For instance, a big disadvantage of a portable unit is that it only cleans the air in the immediate area of the air cleaner!
Having said that, better quality air cleaners incorporate a HEPA or HEPA-type filter. HEPA (High Efficiency Particulate Air) filters have the capability of trapping extremely small particles, including those less than 1 micron in diameter. However, the HEPA filter is expensive, sometimes costing half as much as the entire appliance each time you replace the filter. So, you need to consider your ongoing costs as well as the up-front costs of your air cleaner filter system.
In addition, if your allergies are severe you should be considering a 'whole house' solution. You can get HEPA filters that will work in conjunction with your heating and air conditioning system. Keep in mind that these systems are only filtering the air when your furnace or air conditioner is on and running! The air must be moving through the filter in order to work, therefore the fan for your heating or air conditioning must be on.
Note that true HEPA filters must meet precise standards relating to the percentage of specific particles captured in a laboratory test. Some filters, described as HEPA-type filters are now being sold. These contain a filter made with HEPA materials, but do not quite meet the efficiency standards of true HEPA filters.
Be sure to use allergy-friendly cleaners and vacuum cleaners to supplement any air filtration approach.