A claim on a territory is an assertion by a particular country that the territory, for some particular reason, rightfully belongs to that nation. Claims give a casus belli to the country that has with the claim and give access to the conquest war goal, which makes claims very important for countries seeking to expand by 
war. They can be manually fabricated using the Fabricate claim diplomatic action as well as being awarded by a large variety of events, decisions, and missions, and once made are permanent and cannot be revoked except by special events. Any number of countries can have claims on the same territories.
Claims are visible as yellow hatched lines when a particular country is selected in the 
diplomacy map mode.
Common sources
Common, generalizable sources of claims are listed and described below. Note that this is not an exhaustive list; there are many other less common ways to get claims, particularly flavour events and decisions.
Fabricating claims
Claims can be created using the Fabricate Claims diplomatic action, usable on any province with a territory within 
control range (i.e. neighbouring the current country or sharing a nearby coast, if the country also controls a coast) belonging to a country within 
diplomatic range that is not at war with the current country. Fabricating claims has a base cost of 
20 political influence, modified by the fabricate claim cost modifier, and each claim fabrication has a base monthly progress of 4% per month, which can be increased or decreased by 
claim fabrication speed modifiers. The claim will be awarded on all owned territories in the province once progress reaches 100%, for a base claim fabrication time of 25 months (in practice, the contribution for ruler 
charisma means that it is usually somewhat faster, unless the country has very high 
aggressive expansion). A nation cannot start fabricating another claim if it is already fabricating a claim on that same country, but a country can fabricate on as many different countries concurrently as it can afford.
If a country ceases to control any territories inside the province that claims are being fabricated on, the fabrication process will end immediately with no claim being awarded or political influence refunded. While claims cannot be fabricated on a country that the current country is already at war with, any existing claim fabrications will continue. This means that for larger nations who have too many provinces to fabricate claims on in a reasonable amount of time, it is possible to start fabrication of a claim the month before declaring war, finish fabricating while the claim is ongoing, and then have it available once the peace treaty is being negotiated. Truces do not block claim fabrication; therefore, after attacking a large power, it is useful to use the years of truce to fabricate claims in order to gain a casus belli and reduced 
war score cost and aggressive expansion in the next war.
It possible to fabricate claims on a province even if the current country has a claim on some of the territories in that province belonging to the target country, as long as the target country owns at least one unclaimed territory in the target province. There is no cost or time discount, however, for fabricating claims on already partially claimed provinces.
Nation formation
Most nation formation decisions grant claims on the regions and provinces associated with the particular nation that was formed. Often the forming nation will already control much of the territory in order to meet the requirements for creating the formable in the first place, but the claims granted can still be useful for mopping up remaining territories. One notable exception is 
贵霜, who only requires controlling a few provinces in Sogdiana and Himalaya yet grant claims extending all the way down to Gandhara.
Missions
The generic conquest mission grants claims on up to six provinces in the targeted region after the first task is completed with no resource cost, which can include claims on non-neighbouring provinces that may be impossible to fabricate normally. Combined with the various bonuses from conquering the target provinces, it is rather advantageous to use the generic conquest mission if the target lines up with the country's ambitions. The main drawbacks of the generic conquest mission is that the random target selection means that it can be difficult to get the mission directed at the desired target - as the 
stability cost of aborting the mission makes it impractical to continually reload it - once the nation becomes large and there are many possible neighbouring target regions, and that only three of the province claims are granted upfront, which is typically significantly below a nation's conquest capacity in a single war by the midgame.
Many of the country or culture specific mission trees also give various claims, particularly on territory that is needed to complete the subsequent tasks, though this by no means applies to all mission tasks.
War Council
The 
Summon war council government interaction can be used every 10 years to gain a claim on a province belonging to a neighbouring country. This claim does not come at the cost of political influence and is given immediately - particularly useful at the start of the game - though the potential significant 
loyalty costs, the somewhat long cooldown, and the limited selection of targets possible when summoning the council (particularly as interaction may offer claims on provinces the country already has claims on) generally limits its usefulness compared to fabricating claims.
Effects
War goal
Having a claim on a territory that another country controls gives a casus belli against that country and is the only general way to get access to the conquest war goal, which reduces the warscore cost of all demanded territories by 25% for the attacker, even if they are not actually claimed. This modifier is separate from the usual warscore cost modifier and is applied at the very end.
Territorial demands
A claim on a territory gives -50% aggressive expansion impact and -33% warscore cost when annexing it through a peace treaty or occupation in a Imperial Challenge-type war, which makes it highly desirable to have or make claims on conquest targets, particularly large and populous provinces where aggressive expansion and warscore cost are the highest. Note that the warscore cost reduction is separate from and stacks with the warscore cost reduction from using the conquest war goal.
Releasing nations
When a country is 
released in a peace treaty, they will take with them all territories that they have a claim on belonging to the defeated country. This means that it is possible for a country to be released with territories that they have never owned, particularly common with smaller states that have taken the generic conquest mission and gained claims on neighbouring territories and provinces that they were never able to press.
Opinion effects
Claims do not directly affect the opinion of the country that holds the claims territory, but as claims give a permanent casus belli a country that claims another country's territory will suffer the 
-25 Casus Belli on us opinion malus.