Its could be that you upload Contact records from dataLoader so suppose you are goign to upload 50 contacts records then your partner_member__c records would be 50*4 approximately which is 200 .

Now if you makeing update statement inside the loop it will hit the limit .
So you have to change the approach .

here is the code would work fine !

Trigger updatepartnermember on Contact(after update,after insert ){

Map<Id,List<partner_member__c>> mapContactIdAndPartner = new Map<Id,List<partner_member__c>>();
for(partner_member__c obj :[select Id,Name from partner_member__c where Contact__c in : trigger.newmap.keyset()]){
if(mapContactIdAndPartner.get(obj.Contact__c) == null){
mapContactIdAndPartner.put(obj.Contact__c,new List<partner_member__c>());
}
mapContactIdAndPartner.get(obj.Contact__c).add(obj);
}

List<partner_member__c> lstToUpdate = new List<partner_member__c>();
for(Contact c : trigger.new){
List<partner_member__c> partner_member__lst = mapContactIdAndPartner.get(c.Id);
for(partner_member__c obj :partner_member__lst){
//your logic here
lstToUpdate.add(obj);
}
}

update lstToUpdate;
}

Got bored, so thought I'd pick a fight.

Having a closer look at your code, I would like to recommend a few things.

Trigger updatepartnermember on Contact(after update,after insert ){
	Map<Id,List<partner_member__c>> mapContactIdAndPartner = new Map<Id,List<partner_member__c>>();
	// when you want ID of Sobjects you can just as well pass the SObject list.
	// so instead of giving Contact__c in : trigger.newmap.keyset() , you could as well just give Contact__c in : trigger.new, I prefer trigger.new - it looks better.
	for(partner_member__c obj :[select Id,Name from partner_member__c where Contact__c in : trigger.newmap.keyset()]){
		if(mapContactIdAndPartner.get(obj.Contact__c) == null){
			mapContactIdAndPartner.put(obj.Contact__c,new List<partner_member__c>());
		}
		mapContactIdAndPartner.get(obj.Contact__c).add(obj);
	}

	List<partner_member__c> lstToUpdate = new List<partner_member__c>();
	// why go for another loop here ??
	// what you are trying to do is to iterate throught he partner_member__c records, could have done it in the previous loop itself
	for(Contact c : trigger.new){
		List<partner_member__c> partner_member__lst = mapContactIdAndPartner.get(c.Id);
		// nested loops - bad for coding standards
		for(partner_member__c obj :partner_member__lst){
			//your logic here
			lstToUpdate.add(obj);
		}
	}

	// unguarded update
	update lstToUpdate;
}

A simplified version (??? not as simple as the first one) which lets you do some field mappings, (which is almost always the case) is given below

Trigger updatepartnermember on Contact(after update,after insert ){
	List<partner_member__c> lstToUpdate = new List<partner_member__c>();
	for(partner_member__c obj :[select fieldsNeeded from partner_member__c where Contact__c != null and Contact__c in : trigger.new]){
		obj.Field1__c = 'some value';
		lstToUpdate.add(obj);
	}

	if(!lstToUpdate.isEmpty())
		update lstToUpdate;

}